Single-Trajectory Gibbs Sampling for Non-Commuting Observables

Single-T ra jectory Gibbs Sampling for Non-Comm uting Observ ables Hongrui Chen ∗ 1 , Jiaqing Jiang † 2 , Bo wen Li ‡ 3 , and Lexing Ying § 1,4 1 Departmen t of Mathematics, Stanford Universit y , Stanford, CA 94305, USA 2 Simons Institute for the Theory of Computing, Univ ersity of California, Berk eley , Berk eley , CA 94720, USA 3 Departmen t of Mathematics, City Univ ersit y of Hong Kong, Kowloon T ong, Hong Kong SAR 4 Institute for Computational and Mathematical Engineering, Stanford Universit y , Stanford, CA 94305, USA Marc h 24, 2026 Abstract Estimating thermal expectation v alues of quan tum many-bo dy systems is a cen tral challenge in physics, chemistry , and materials science. Standard quantum Gibbs sampling proto cols ad- dress this task by preparing the Gibbs state from scratch after ev ery measuremen t, incurring a full mixing-time cost at each step. Recen t adv ances in single-tra jectory Gibbs sampling [ JLL26 ] substan tially reduce this o verhead: once stationarity is reac hed, measurements can b e collected along a single tra jectory without re-thermalizing, provided the measurement channel preserves the Gibbs ensem ble. Ho wev er, explicit constructions of suc h non-destructiv e measuremen ts ha ve b een limited primarily to observ ables that commute with the Hamiltonian. In this w ork, we fundamen tally extend the single-tra jectory framework to arbitrary , non-comm uting observ ables. W e pro vide tw o measuremen t constructions that extract measuremen t information without fully destro ying the Gibbs state, thereb y eliminating the need for full re-mixing b et ween samples. First, w e construct a measuremen t that satisﬁes exact detailed balance. This ensures the system remains in equilibrium throughout the tra jectory , allo wing measuremen t outcomes to decorrelate in an auto correlation time that could be signiﬁcan tly shorter than the global mixing time. Second, assuming the underlying quantum Gibbs sampler has a positive sp ectral gap, we design a simpliﬁed measurement sc heme that ensures the p ost-selected state serves as a w arm start for rapid re-mixing. This approach successfully decouples the resampling cost from the global mixing time. Both measurement sc hemes admit eﬃcien t quan tum circuit implemen ta- tions, requiring only p olylogarithmic Hamiltonian sim ulation time. ∗ hongrui@stanford.edu † jiaqingjiang95@gmail.com ‡ bowen.li@cityu.edu.hk § lexing@stanford.edu 1 Con ten ts 1 In tro duction 2 1.1 Ov erview of our Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Related W orks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Preliminaries 7 2.1 Quan tum Detailed Balance and the Sp ectral Gap . . . . . . . . . . . . . . . . . . . . 8 2.2 Smo othing Non-comm uting Observ ables . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Measuremen ts via Detailed balanced Channels 11 3.1 Construction of the Jump Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Construction of the Rejection Branch . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 The Single-T ra jectory Proto col . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Measuremen t-Remixing Strategy: W arm-Start in χ 2 -Div ergence 17 4.1 Construction of the Measurement Channel . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 The W arm-Start Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Ov erall Complexit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 A The Measuremen t Process and the Outcome Statistics 22 A.1 Stabilit y Analysis of Implementation Errors . . . . . . . . . . . . . . . . . . . . . . . 23 A.2 The Auto correlation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 B Implementation of the Measurement Channels 27 B.1 Blo c k enco ding of A f and A ˜ f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 B.2 Blo c k enco ding of c p I + uA f and c q I + uA ˜ f . . . . . . . . . . . . . . . . . . . . . 32 B.2.1 The QSVT construction for real ﬁlters . . . . . . . . . . . . . . . . . . . . . . 33 B.2.2 The LCU construction for general ﬁlters . . . . . . . . . . . . . . . . . . . . . 34 B.3 Appro ximate Implemen tation of the POVM . . . . . . . . . . . . . . . . . . . . . . 35 C Implemen tation of the sup eroperator R 38 1 In tro duction Estimating thermal exp ectation v alues is a central task in quantum physics and quan tum chemistry , underpinning our theoretical understanding of quan tum many-bo dy systems at ﬁnite temperatures. A t inv erse temp erature β , a quantum system in thermal equilibrium is gov erned by the Gibbs state σ β = e − β H / Z , where H is the Hamiltonian and Z = T r( e − β H ) is the partition function. Key prop erties of the system, suc h as energy , magnetization, and correlation functions, are enco ded in the thermal exp ectation v alue, deﬁned as ⟨ A ⟩ σ β = T r( σ β A ) for the resp ectiv e observ able A . Computing thermal exp ectation v alues for general quantum systems is intractable on classical computers due to the sign problem and the exp onen tial growth of the Hilb ert space, whic h motiv ates the developmen t of quan tum algorithms. Recen t adv ances in quantum Gibbs sampling [ CKBG25 , CK G25 , DLL25 , JI24 , TO V + 11 ] pro- vide a foundation for estimating thermal exp ectation v alues on quan tum computers, op ening the 2 p ossibilit y of inv estigating quantum man y-b o dy systems b ey ond classical computational limits. The quan tum Gibbs sampling algorithm, whic h serv es as a quantum analogue of classical Monte Carlo metho ds, can driv e any initial state to the Gibbs state within a timescale known as the mixing time t mix . T o estimate thermal exp ectation v alues to precision ϵ , it suﬃces to prepare O (1 /ϵ 2 ) Gibbs states and perform measuremen ts, resulting in a total Gibbs sampling time of O ( t mix /ϵ 2 ). While this approach pro vides a viable method for estimating thermal expectation v alues, its O ( t mix /ϵ 2 ) scaling can b e prohibitiv ely large, motiv ating the dev elopment of tec hniques that further reduce the sampling cost. T o address these c hallenges, [ JLL26 ] introduced a new paradigm for thermal exp ectation esti- mation, demonstrating that the sampling cost can b e substan tially reduced using a single Gibbs- sampling tra jectory . Their algorithm ﬁrst runs the Gibbs sampler for roughly one mixing time to bring the giv en initial state close to the Gibbs state, then p erforms coheren t measuremen ts at regular in terv als to collect samples. The k ey observ ation is that if the measurement c hannel M satisﬁes detailed balance and hence preserv es the Gibbs state, i.e., M ( σ β ) = σ β , then the chain remains in the Gibbs ensem ble during the sampling stage. Consequen tly , once the c hain has reached stationarit y , consecutive samples b ecome eﬀectively indep enden t on a timescale muc h shorter than the mixing time. This timescale is captured b y the auto c orr elation time t aut , whic h quan tiﬁes the rate of decorrelation in stationarity . As a result, to estimate thermal exp ectation v alues to ac- curacy ϵ , the total Gibbs sampling cost is t mix + O ( t aut /ϵ 2 ), whic h is substantially more eﬃcient than O ( t mix /ϵ 2 ) whenever t aut ≪ t mix . [ JLL26 , Section 1.3] also provided examples in whic h t aut is signiﬁcan tly smaller than t mix . More generally , they sho wed that the auto correlation time t aut is upp er b ounded b y the inv erse sp ectral gap of the quantum Gibbs sampler, whic h in principle could b e smaller than the mixing time b y a factor prop ortional to the system size. As men tioned, the key ingredien t enabling the sampling reduction in [ JLL26 ] is the non- destructiv e measurement of Gibbs states, meaning that it preserv es the Gibbs state ensem ble ( M ( σ β ) = σ β ). F or observ ables that comm ute with the Hamiltonian, they explicitly construct a non-destructiv e measurement with only logarithmic ov erhead via Gaussian-ﬁltered quantum phase estimation [ Mou19 ]. F or general observ ables, under additional assumptions, they show ed that the w eighted op erator F ourier transform [ CK G25 ] can b e used to mitigate measurement-induced distur- bance. Ho wev er, the construction of non-destructive measuremen ts for general observ ables remains an op en problem. In this w ork, w e close this gap by constructing non-destructiv e measuremen t channels (satisfying detailed balance) for general observ ables that may not comm ute with the Hamiltonian. As a consequence, our construction fully extends the single-tra jectory framew ork to the non-comm uting setting. Theorem 1 (Exact detailed-balance measurement (informal v ersion of Theorem 6 )) . L et σ β = e − β H / T r( e − β H ) b e the Gibbs state of a Hamiltonian H at inverse temp er atur e β . Consider any Hermitian observable A with ∥ A ∥ ≤ 1 . One c an c onstruct a quantum channel M , • M satisﬁes detaile d b alanc e with r esp e ct to σ β , and henc e ﬁxes the Gibbs state M ( σ β ) = σ β . • The outc omes of M yield an unbiase d estimator of T r( σ β A ) of ˜ O (1) varianc e. • A n ϵ -appr oximate implementation of M c an b e r e alize d in ˜ O (1) queries to blo ck enc o ding of A , and ˜ O ( β ) (c ontr ol le d) Hamiltonian simulation time of H , and ˜ O (1) ancil la qubits. 3 wher e ˜ O suppr esses p olylo garithmic factors in β , ∥ H ∥ , and ϵ . Combining the c onstructe d me a- sur ement with the single-tr aje ctory Gibbs sampling appr o ach [ JLL26 ], one c an estimate T r( σ β A ) to pr e cision ϵ with failur e pr ob ability η using a total Gibbs sampling time of t mix + O ( t aut / ( ϵ 2 η )) , wher e t aut denotes the c orr esp onding auto c orr elation time and is b ounde d by the inverse sp e ctr al gap (up to p olylo garithmic factor) of the Gibbs sampling channel. Theorem 1 implies that, for general observ ables, our constructed measurement combined with the single-tra jectory Gibbs sampling approach [ JLL26 ] allows one to obtain eﬀectively independent samples every autoc or relation time rather than per mixing time, leading to a signiﬁcan t reduction in the Gibbs sampling cost for thermal exp ectation estimation. As illustrated in the examples of [ JLL26 ], in many scenarios the auto correlation time t aut can b e muc h shorter than the mixing time (potentially b y a sub-exponential factor), since it lev erages a w arm start from the previous measuremen t and depends on the observ able of interest. More precisely , t aut is upper b ounded by the in verse sp ectral gap λ − 1 , whereas the mixing time may carry the additional factor log ( σ − 1 β , min ), where σ β , min denotes the smallest eigenv alue of σ β that could b e exp onen tially small in the n umber of qubits n . The single-tra jectory approach a voids this prefactor en tirely in the p er-sample cost. In addition to Theorem 1 , w e also construct a simpler measurement that forgo es the detailed- balance prop ert y but instead guaran tees that the p ost-selected state remains a w arm start for the Gibbs sampler. Under a spectral gap assumption, this ensures that each remixing step costs only in verse-spectral-gap time rather than the full mixing time. Theorem 2 (Informal v ersion of Theorem 10 ) . L et σ β , H , and A b e as in The or em 1 . Supp ose that N is a quantum Gibbs sampling channel with sp e ctr al gap λ > 0 . One c an c onstruct a me asur ement channel M such that: • The p ost-sele cte d state is a warm start for N such that r e aching the Gibbs state again within ϵ ac cur acy r e quir es only O ( λ − 1 log(1 /ϵ )) Gibbs sampling time. • The outc omes of M yield an unbiase d estimator of T r( σ β A ) of O (1) varianc e. Conse quently, one c an estimate T r( σ β A ) to pr e cision ϵ with failur e pr ob ability η using a total Gibbs sampling time of t mix + O ( λ − 1 log(1 /η ) /ϵ 2 ) . In the following section, we giv e an o verview of the construction of our t wo measurement c hannels: an exact detailed-balance c hannel, and a simpliﬁed c hannel that do es not preserve detailed balance but guaran tees a constan t χ 2 -div ergence w arm start. Readers in terested in further details on the single-tra jectory approach ma y refer to Section 3.3 or [ JLL26 ]. 1.1 Ov erview of our Construction Recall that w e use σ β ∝ exp( − β H ) to denote the Gibbs state with respect to Hamiltonian H and in verse temp erature β . Consider an observ able A with ∥ A ∥ ≤ 1. Our goal is to design measure- men t channels that simultaneously extract an unbiased estimator of the thermal expectation v alue T r( σ β A ) and lea v e the p ost-measuremen t state in a con trolled proximit y to σ β , either b y exactly preserving the Gibbs state via detailed balance, or b y guaran teeing a b ounded χ 2 -div ergence w arm start that enables fast remixing. T o build some in tuition, note that for observ ables commuting with H , i.e., [ A, H ] = 0, one can measure A without disturbing the Gibbs state ensem ble, for instance via pro jectiv e measurement 4 in the energy eigenbasis or quantum phase estimation. F or general non-comm uting observ ables, ho wev er, pro jectiv e measurement can signiﬁcan tly p erturb the Gibbs state, motiv ating the need for more carefully designed measurement c hannels. A natural approac h to mitigating measuremen t disturbance, also discussed in [ JLL26 , CK25 ], is to smooth the observ able A via the op erator F ourier transform, A f = Z + ∞ −∞ f ( t ) e iH t Ae − iH t dt, (1) where f ( t ) = 1 √ 2 π τ 2 exp  − t 2 / 2 τ 2  is a Gaussian with width parameter τ . The smo othed observ able A f has several desirable prop erties [ JLL26 ]. First, it preserves the thermal exp ectation v alue, T r( A f σ β ) = T r( Aσ β ), so one can equiv alen tly measure A f in place of A . Second, when τ is large, A f appro ximately comm utes with σ β , so its measuremen t incurs only a sligh t disturbance to the Gibbs state. Ho wev er, the implementation cost of the measurement of A f scales as ˜ O ( τ ) while ensuring approximate comm utation with H in the non-commuting case generally requires τ to b e large leading to signiﬁcant o v erhead. Besides, the resulting measuremen t only appro ximately preserv es th e Gibbs state and do es not satisfy exact detailed balance, so the single-tra jectory sample complexit y theorem of [ JLL26 ] do es not directly apply . W e presen t tw o constructions that address these limitations in diﬀeren t wa ys. In the ﬁrst construction, we show that it is possible to construct a measuremen t that s ati sﬁes exact detailed balance, based on A f with a mo derate τ = ˜ O ( β ). In the second construction, we develop a measuremen t that p erturbs the Gibbs state but guarantees a constant χ 2 -div ergence w arm start, enabling fast remixing back to the Gibbs state. W e begin with a brief review of relev an t concepts; see Section 2 for details. A quan tum measure- men t channel is describ ed by its Kraus op erators { K a } a , satisfying the trace-preserving condition P a K † a K a = I , under which a state ρ is mapp ed to P a K a ρK † a . A quan tum channel (or more generally , a c ompletely p ositive map) is said to satisfy the Kub o–Martin–Sc hwinger (KMS) de- tailed balance condition with respect to σ β if it is self-adjoin t under the KMS inner pro duct, or equiv alen tly , if it satisﬁes X a σ − 1 / 2 β K a ρK † a σ − 1 / 2 β = X a K † a σ − 1 / 2 β ρ σ − 1 / 2 β K a . Setting ρ = σ β and using the trace-preserving prop erty , one v eriﬁes that any KMS detailed-balanced c hannel necessarily ﬁxes the Gibbs state: M ( σ β ) = σ β . Exact Detailed-Balance Measuremen t for Non-Comm uting Observ ables. Our ﬁrst con- struction is a measurement c hannel with Kraus op erators K 1 = c p I + uA f , K 2 = c q I + uA ˜ f , K = q √ σ β ( I − K † 1 K 1 − K † 2 K 2 ) √ σ β σ − 1 / 2 β . (2) W e now explain the role of each comp onen t and why the construction satisﬁes KMS detailed balance; its estimation prop erties and implemen tation cost are discussed subsequently . A more tec hnical treatmen t is pro vided in Section 3 . Here f ( t ) = 1 √ 2 π τ 2 exp  − t 2 / 2 τ 2  is the Gaussian ﬁlter with width τ , and ˜ f ( t ) := f ( t − iβ / 2) is its imaginary-time shift by iβ / 2. The shifted ﬁlter is c hosen precisely to enforce the algebraic relation K 2 = σ 1 / 2 β K † 1 σ − 1 / 2 β , whic h, as sho wn in Lemma 7 , is suﬃcient to guarantee that the completely 5 p ositiv e (CP) map T ′ [ ρ ] = K 1 ρK † 1 + K 2 ρK † 2 satisﬁes KMS detailed balance with resp ect to σ β . Since T ′ is not trace-preserving, it do es not by itself deﬁne a v alid quan tum channel. F ollo wing the discrete-time quantum Metrop olis–Hastings framework of [ GCDK26 ], we app end a rejection branc h with Kraus op erator K that restores the trace-preserving property while main taining the exact KMS detailed balance. The parameters c and u are rescaling factors to ensure the square ro ot function in the deﬁnition ( 2 ) of Kraus op erators is well-deﬁned. T o a void confusion, we note that since ˜ f is not a real symmetric function, the op erator A ˜ f is not Hermitian. The square ro ot in K 2 is deﬁned via an absolutely con vergen t T aylor expansion. The thermal exp ectation v alue is naturally extracted from the measuremen t outcomes as follows. Assign outcome v alues z = 1 to branc hes K 1 and K 2 , and z = 0 to the rejection branc h K . When the system is in the Gibbs state σ β , the exp ected outcome E [ z ] equals the total probabilit y of the non-rejected branches: T r( K 1 σ β K † 1 ) + T r( K 2 σ β K † 2 ) = 2 c 2 (1 + u T r( A f σ β )) = 2 c 2 (1 + u T r( Aσ β )) , (3) where the ﬁrst equality uses K 2 = σ 1 / 2 β K † 1 σ − 1 / 2 β and the second uses the fact that A f preserv es the thermal exp ectation v alue, T r( A f σ β ) = T r( Aσ β ). It follo ws that the random v ariable v := z 2 c 2 u − 1 u (4) is an un biased estimator of T r( Aσ β ), with v ariance of order ( cu ) − 2 . The performance of the constructed measuremen t is go v erned by the parameters c and u . A crucial prop ert y of the smo othed operators is that the norms ∥ A f ∥ and ∥ A ˜ f ∥ remain O (1) when τ = Ω( β ). Consequen tly , it suﬃces to set τ = Θ( β ) and c, u = e O (1) to ensure that u ∥ A f ∥ and u ∥ A ˜ f ∥ are b ounded aw a y from 1, so that the matrix square ro ots in the Kraus op erators are well-deﬁned with rapidly conv ergen t T aylor expansions. Based on this choice of parameters and the techniques for implementing the rejection op erator K from [ GCDK26 ], we implemen t the measuremen t to precision ϵ using e O (1) queries to a blo ck enco ding of A , e O ( β ) con trolled Hamiltonian simulation time, and e O (1) ancilla qubits, via the linear combination of unitaries and quan tum singular v alue transformation [ GSL W19 ]. Measuremen t-Remix Strategy v ia W arm Starts in χ 2 -Div ergence. Our second construc- tion is simpler. W e deﬁne a tw o-outcome measurement with Kraus op erators K ± = r 1 2 ( I ± uA f ) , (5) where the outcome ± is assigned the v alue z = ± 1. Up on observing outcome ± , the p ost- measuremen t state is ρ ′ ± = K ± ρK † ± /p ± , and the diﬀerence in outcome probabilities p + − p − = u T r( ρA f ) yields an unbiased estimator of T r( σ β A ). Since this construction uses only the real-time smo othed observ able A f , requiring no imaginary-time shift and no rejection branc h, it is consider- ably simpler to implement than the abov e exact detailed-balance channel. Ho wev er, this measurement channel do es not satisfy exact detailed balance, and the post- measuremen t state is disturb ed. The k ey result is that the disturbance remains controlled: with τ = Ω( β ), the post-selected state ρ ′ ± satisﬁes χ 2 ( ρ ′ ± , σ β ) = O (1), providing a w arm start for the subsequen t Gibbs sampling dynamics. Under a spectral gap λ > 0, this guarantees that the state ρ ′ ± returns to ϵ -proximit y of σ β in remixing time O ( λ − 1 log(1 /ϵ )), eliminating the O ( n ) o v erhead from log ( σ − 1 β , min ) that appears in the general mixing time estimate. W e refer to Section 4 for the precise statement and pro of. 6 1.2 Related W orks This work studies the problem of measuring a Gibbs state non-destructively: either preserving it exactly via detailed balance, or p erturbing it in a con trolled wa y that allo ws eﬃcien t recov ery . F or non-destructive measurements of Gibbs states, the most relev an t prior work is [ JLL26 ], whic h constructs non-destructiv e measuremen ts for observ ables commuting with H , incurring only logarithmic o v erhead. F or more general observ ables, [ JLL26 ] emplo ys the weigh ted operator F ourier transform to construct a measuremen t that only sligh tly disturbs the Gibbs state, under additional assumptions. Our Theorem 1 generalizes the result of [ JLL26 ] to arbitrary observ ables. While the o verhead is larger than in the comm uting case, it remains modest: scaling logarithmically in both the desired precision and the Hamiltonian evolution time, and linear in β Another approach to near-non-destructiv e measurement is weak measurement [ V ai14 ], which, how ev er, can substantially increase the sample complexity for estimating thermal exp ectation v alues to a given precision. Non-destructiv e measuremen ts ha ve b een more extensively studied in the context of ground states. F or gapp ed systems in particular, such measuremen ts can b e realized via a v ariet y of tec hniques, including the weigh ted op erator F ourier transform [ CK25 ], weak measuremen t [ V ai14 ], and the Marriott–W atrous rewinding technique [ MW05 , FGH + 10 ]. A detailed survey of these metho ds and a comparison of their implementation costs can b e found in [ CK25 ]. Another approach to non-destructive measurement is based on recov ery maps: one ﬁrst p erforms a measuremen t on the Gibbs state and then applies a recov ery procedure to bring the disturb ed state bac k to the Gibbs state. In particular, based on lo cal Mark ov prop erties, it has b een shown that the disturbance caused by measuring lo cal observ ables can b e reco v ered by quasi-lo cal quantum c hannels [ CR25 , KK25 ]. As presented abov e, the design of non-destructive measuremen ts is closely related to the de- sign of quantum Gibbs sampling algorithms. Our constructions build on t wo key ingredients: the smoothing technique via the w eighted op erator F ourier transform [ CK G25 ] and the discrete- time quantum Metrop olis–Hastings framew ork [ GCDK26 ]. T echniques and insights from other Gibbs sampling algorithms [ JI24 , DLL25 , R WW23 , TO V + 11 , DZPL25 ] and related structural re- sults [ BCV25 ] may further inform the design of new measurement proto cols with simpler con- structions or reduced implementation costs. Conv ersely , a fundamen tally diﬀerent approac h to non-destructiv e measuremen t for general observ ables would itself yield a new Gibbs sampling algo- rithm, since any suc h measurement corresp onds to a quantum c hannel admitting the Gibbs state as a ﬁxed p oint. 2 Preliminaries In this section, we introduce some basic concepts used throughout the pap er, particularly the quan tum Gibbs sampler, the KMS detailed balance condition, the spectral gap, and Heisen b erg- picture smo othed observ ables via the operator F ourier transform. In this w ork, we consider a ﬁnite-dimensional quan tum system asso ciated with a Hilb ert space H ∼ = C d , where d = 2 n and n is the n umber of qubits. Let B ( H ) ∼ = C d × d denote the algebra of b ounded linear operators on H , and let D ( H ) b e the subset of quan tum states. Given a system Hamiltonian H ∈ B ( H ) and in verse temperature β > 0, the quan tum Gibbs (thermal) state is giv en b y σ β = e − β H / T r( e − β H ). A standard approach to preparing quantum thermal states is quantum Gibbs sampling: one designs a primitive quantum channel N , eﬃciently simulable on a quan tum computer, that admits 7 the Gibbs state σ β as its unique ﬁxed p oin t [ CKBG25 , DLL25 , CK G25 , GCDK26 ], where lim k →∞ N k ( ρ ) = σ β for every initial state ρ. (6) The conv ergence rate of N is quan tiﬁed by the mixing time , t mix ( ϵ ) := min n k ≥ 0    sup ρ ∈D ( H ) ∥N k ( ρ ) − σ β ∥ 1 ≤ ϵ o , (7) where ∥ · ∥ 1 denotes the trace norm. Our ob jective is to design quantum algorithms for estimating the thermal exp ectation v alue ⟨ A ⟩ σ β := T r( σ β A ) of a target Hermitian observ able A to additiv e error ϵ . T o this end, as preview ed in Theorems 1 and 2 , we develop tw o measurement proto cols for Gibbs states and analyze how each reduces the o verall sampling cost. 2.1 Quan tum Detailed Balance and the Sp ectral Gap One of the key features of existing quantum Gibbs samplers is the KMS detailed balance condi- tion, which facilitates both quantum implementation [ CK G25 , DLL25 , GCDK26 ] and conv ergence analysis [ RF A25 , RSF A26 ]. F or any X , Y ∈ B ( H ), the KMS inner product is deﬁned by ⟨ X , Y ⟩ σ β , 1 / 2 := T r  X † σ 1 / 2 β Y σ 1 / 2 β  . (8) A sup erop erator T : B ( H ) → B ( H ) is called a quantum channel if it is completely p ositive and trace-preserving, and we denote b y T † its adjoint with respect to the Hilbert–Schmidt inner product ⟨ X , Y ⟩ HS := T r( X † Y ). Deﬁnition 3 (KMS detailed balance) . A sup erop erator T : B ( H ) → B ( H ) satisﬁes the KMS detailed balance condition with respect to σ β if T † is self-adjoin t under the KMS inner pro duct, i.e., for all X , Y ∈ B ( H ), ⟨ X , T † ( Y ) ⟩ σ β , 1 / 2 = ⟨T † ( X ) , Y ⟩ σ β , 1 / 2 . (9) An y σ β -KMS detailed balanced quan tum channel T necessarily admits σ β as a ﬁxed p oin t, T ( σ β ) = σ β . T o b ound the mixing time, a standard approac h is to analyze the con v ergence of the iterates T k via the χ 2 -div ergence. W e ﬁrst in tro duce the weigh ted norm ∥ X ∥ 2 σ β , − 1 / 2 := T r  σ − 1 / 2 β X † σ − 1 / 2 β X  , X ∈ B ( H ) , (10) whic h is closely related to the family of monotone Riemannian metrics in quan tum informa- tion [ PS96 , LR99 ]; see [ CLL Y25 , Section 2] for the precise connection. The χ 2 -div ergence betw een a state ρ and the Gibbs state σ β is then deﬁned by χ 2 ( ρ, σ β ) := ∥ ρ − σ β ∥ 2 σ β , − 1 / 2 = T r  σ − 1 / 2 β ( ρ − σ β ) σ − 1 / 2 β ( ρ − σ β )  . (11) According to [ TKR + 10 ], the decay of χ 2 ( N k ( ρ ) , σ β ) for the discrete-time quantum Mark ov c hain {N k } k ≥ 0 is controlled b y the sp ectral gap. Deﬁnition 4. Let T b e a primitiv e detailed-balanced quantum channel with in v arian t state σ β , and assume that its sp ectrum is con tained in [0 , 1]. The sp e ctr al gap of T is deﬁned by gap( T ) := 1 − max X  =0 ⟨ X,I ⟩ σ β , 1 / 2 =0 ⟨ X , T † ( X ) ⟩ σ β , 1 / 2 ⟨ X , X ⟩ σ β , 1 / 2 . 8 A strictly p ositiv e sp ectral gap, λ := gap( T ) > 0, implies the exp onen tial deca y of the χ 2 - div ergence under iteration: χ 2 ( T k ( ρ ) , σ β ) ≤ (1 − λ ) 2 k χ 2 ( ρ, σ β ) , whic h giv es, using ∥ ρ − σ β ∥ 2 1 ≤ χ 2 ( ρ, σ β ), ∥T k ( ρ ) − σ β ∥ 1 ≤ q χ 2 ( T k ( ρ ) , σ β ) ≤ (1 − λ ) k q χ 2 ( ρ, σ β ) . F urthermore, the mixing time scales as t mix ( ϵ ) = O  1 λ log 1 ϵ σ β , min  , where σ β , min denotes the smallest eigenv alue of σ β . 2.2 Smo othing Non-commuting Observ ables A central diﬃculty in extending [ JLL26 ] to estimating the thermal exp ectation ⟨ A ⟩ σ β for observ ables A , which do not commute with the Hamiltonian H , is that direct pro jective measuremen ts severely disturb the Gibbs state, and the resulting p ost-measuremen t states ma y not av erage back to σ β . T o extract measurement information while main taining a controlled disturbance on σ β , w e use a Heisen b erg-picture smo othed observ able. F ollo wing [ CKG25 , DLL25 , JLL26 , CK25 ], for any Hermitian observ able A , w e deﬁne the ﬁltered (p ossibly non-Hermitian) op erator A f asso ciated with an L 1 -in tegrable ﬁlter function f : R → C b y A f := Z ∞ −∞ f ( t ) e iH t Ae − iH t d t. (12) W e assume that f is normalized, i.e., Z ∞ −∞ f ( t ) d t = 1 , whic h, noting [ σ β , e iH t ] = 0, implies that A f preserv es the thermal exp ectation v alue of A : T r( σ β A f ) = Z ∞ −∞ f ( t ) T r  e − iH t σ β e iH t A  d t = Z ∞ −∞ f ( t ) T r( σ β A ) d t = T r( σ β A ) . (13) This prop erty is crucial for constructing an unbiased estimator of ⟨ A ⟩ σ β . A technically essential ingredient in our detailed-balanced measurement sc heme (Section 3 ) and warm-start analysis (Section 4 ) is the controlled behavior of observ ables under imaginary- time evolution. Although for any ﬁxed ﬁnite-dimensional system, the operator e sH Ae − sH remains b ounded for ev ery ﬁnite s , its norm can still grow rapidly (see further discussion in [ CR25 , BC26 ]). In the thermo dynamic limit, lo cal observ ables may even lose b oundedness or quasi-lo cal control at ﬁnite imaginary time [ Bou15 , PGPH23 ]. The purp ose of ﬁlter smo othing is precisely to tame this gro wth through a suitably c hosen ﬁlter function. 9 Lemma 5 (Smo othed observ able) . F or a given s ∈ R , supp ose that the F ourier tr ansform ˆ f ( ω ) = R ∞ −∞ f ( t ) e − iω t d t of the ﬁlter function satisﬁes | ˆ f ( ω ) | ≤ C e − γ | ω | , for some c onstant C > 0 and γ > | s | . (14) Then the imaginary-time evolution of A f is again a ﬁlter e d op er ator: e sH A f e − sH = A ˜ f , (15) wher e ˜ f is the analytic al ly c ontinue d ﬁlter function deﬁne d by ˜ f ( t ) := f ( t + is ) = 1 2 π Z ∞ −∞ ˆ f ( ω ) e − ω s + iω t d ω . If, in addition, ˜ f ∈ L 1 ( R ) , then ∥ A ˜ f ∥ ≤ ∥ A ∥ Z ∞ −∞ | ˜ f ( t ) | d t. (16) In p articular, for the Gaussian ﬁlter f ( t ) = 1 √ 2 π τ 2 e − t 2 / 2 τ 2 , we have ∥ A ˜ f ∥ ≤ e s 2 / 2 τ 2 ∥ A ∥ . Pr o of. Let H = P k E k | k ⟩⟨ k | b e the sp ectral decomp osition of H , where E k are the energy eigen- v alues and | k ⟩ the corresponding energy eigenstates. The matrix elemen ts of A f are ⟨ k | A f | j ⟩ = Z ∞ −∞ f ( t ) e i ( E k − E j ) t ⟨ k | A | j ⟩ d t = ˆ f ( E j − E k ) ⟨ k | A | j ⟩ . Conjugating by e sH and e − sH rescales these elemen ts by e s ( E k − E j ) : ⟨ k | e sH A f e − sH | j ⟩ = e s ( E k − E j ) ˆ f ( E j − E k ) ⟨ k | A | j ⟩ . Since | ˆ f ( ω ) | ≤ C e − γ | ω | b y assumption and γ > | s | , the pro duct e − sω ˆ f ( ω ) is w ell-deﬁned and absolutely integrable. Moreov er, it is b ounded for all ω = E j − E k . Note that ˜ f ( t ) is deﬁned as the in verse F ourier transform of e − sω ˆ f ( ω ). W e hav e ˆ ˜ f ( ω ) = e − sω ˆ f ( ω ), and ⟨ k | e sH A f e − sH | j ⟩ = ˆ ˜ f ( E j − E k ) ⟨ k | A | j ⟩ = ⟨ k | A ˜ f | j ⟩ , that is, e sH A f e − sH = A ˜ f . The norm bound ( 16 ) follows from the triangle inequality applied to the in tegral deﬁnition of A ˜ f : ∥ A ˜ f ∥ ≤ Z ∞ −∞ | ˜ f ( t ) | ∥ e iH t Ae − iH t ∥ d t ≤ ∥ A ∥ Z ∞ −∞ | ˜ f ( t ) | d t, where we used the fact that conjugation b y unitaries preserves the op erator norm. By standard P aley–Wiener type argumen ts, the exp onen tial decay condition ( 14 ) implies that f extends analytically to the strip | Im z | < γ . Thus, Theorem 5 shows that, with a suﬃciently regular ﬁlter function such as a Gaussian, the ﬁltered observ able A f remains well b ehav ed under imaginary-time evolution e sH A f e − sH , in the sense that its norm can b e controlled through the analytically con tinued ﬁlter. This prop erty pla ys an essen tial role in constructing detailed-balanced measuremen t op erators without rapid norm growth in the lo w-temp erature regime (Section 3 ) and in establishing the warm-start property of post-measurement states (Section 4 ). 10 3 Measuremen ts via Detailed balanced Channels In this section, we design a measurement channel satisfying the exact KMS detailed balance condi- tion, and generalizes the single-tra jectory Gibbs sampling approac h [ JLL26 ] to the non-comm uting setting and reducing the sampling cost for estimating thermal expectation v alues. Sp eciﬁcally , we shall construct a c hannel that yields an unbiased estimator of T r( σ β A ) for arbitrary observ ables A not commuting with H , while leaving the Gibbs state σ β in v arian t. The construction follo ws the discrete-time quan tum Metrop olis-Hastings framework of [ GCDK26 ]: the c hannel consists of a jump branch that enco des the desired measurement information and satisﬁes exact detailed balance, and a rejection branch that enforces the trace-preserving prop erty . Sp eciﬁcally , our construction pro ceeds in tw o main steps. First, we design Kraus op erators K 1 and K 2 in ( 20 ) and ( 22 ), based on the smo othed observ ables A f and A ˜ f , whose outcome probabilities enco de T r( σ β A ) and whic h jointly satisfy exact KMS detailed balance (see Theorem 7 ). The imaginary-time shift in the ﬁlter function ˜ f of K 2 is precisely what enforces the detailed balance relation. Since K 1 and K 2 alone do not deﬁne a trace-preserving channel, w e follo w the quan tum Metrop olis–Hastings approach of [ GCDK26 ] and app end a rejection Kraus operator K that restores the trace-preserving prop erty while maintaining exact detailed balance. The resulting measuremen t c hannel M is then applied within the single-tra jectory framew ork of [ JLL26 ], where the auto correlation time t aut go verns the n umber of steps needed b et ween consecutiv e samples to decorrelate; see Section 3.3 b elo w. Without loss of generality , we assume the target Hermitian observ able A satisﬁes ∥ A ∥ ≤ 1. Otherwise, one can replace A b y the rescaled observ able A ′ = A/ ∥ A ∥ , in whic h case, to estimate T r( σ β A ) to additive error ϵ , it suﬃces to estimate T r( σ β A ′ ) to additive error ϵ ′ = ϵ/ ∥ A ∥ . Our main result is the following theorem. Theorem 6. (Detaile d-Balanc e d Me asur ement and Sample Complexity) L et A b e a Hermitian observable with || A || ≤ 1 . Deﬁne the Gaussian ﬁlter f and the asso ciate d shifte d one ˜ f by f ( t ) := 1 √ 2 π τ 2 exp  − t 2 2 τ 2  , ˜ f ( t ) := f  t − i β 2  . F or any tar get ac cur acy ϵ , one c an cho ose the width τ = O ( β + p olylog ( ∥ H ∥ /ϵ )) , the me asur ement str ength u = O (1) and the sc aling c onstant c = O (1 / log ( β ∥ H ∥ )) , such that the me asur ement pr oto c ol deﬁne d by the quantum channel: M [ ρ ] = K 1 ρK † 1 + K 2 ρK † 2 + K ρK † , with Kr aus op er ators: K 1 = c p I + uA f , K 2 = c q I + uA ˜ f , K = q √ σ β ( I − K † 1 K 1 − K † 2 K 2 ) √ σ β σ − 1 / 2 β , satisﬁes the fol lowing pr op erties: • Exact Detaile d Balanc e : The channel M is tr ac e-pr eserving and satisﬁes the KMS detaile d b alanc e c ondition with r esp e ct to σ β . • Eﬃcient Implementation: The channel M , as a quantum instrument, c an b e simulate d with ϵ ′ appr oximation err or (in the sense of Deﬁnition 13 ), using O (p olylog ( β ∥ H ∥ /ϵ ′ )) queries to blo ck enc o ding of A , O ( β p olylog ( ∥ H ∥ /ϵ ′ )) (c ontr ol le d) Hamiltonian simulation time of H , and O (p olylog ( β ∥ H ∥ /ϵ ′ )) ancil la qubits. 11 • The Over al l Complexity. Consider a primitive quantum Gibbs sampling channel N satisfying KMS detaile d b alanc e, e.g. [ CKBG25 , DLL25 , CK G25 , GCDK26 ]. T o estimate T r( σ β A ) to pr e cision ϵ with failur e pr ob ability η , it suﬃc es to pr o c e e d as fol lows. (i)Apply t mix ( η / 3) steps of N to pr ep ar e an initial state ρ burn satisfying | ρ burn − σ β | 1 ≤ η / 3 . (ii)Apply an ϵ ′ -appr oximate implementation of the c omp osite channel M ◦ N ◦ M (in the sense of Deﬁnition 13 ) for T = O  t aut ,T log 2 ( β | H | ) ϵ 2 η  steps with p er-step ac cur acy ϵ ′ = η /T , wher e t aut ,T is the auto c orr elation time deﬁne d in Eq. ( 24 ). In addition, if N has sp e ctrum in [0 , 1] and admits a sp e ctr al gap λ > 0 , the auto c orr elation time is upp er b ounde d by t aut ,T ≤ O (1) λ log 2 ( β | H | ) + 1 2 . In what follo ws, we detail eac h component of this construction and the resulting proto col, build- ing to w ard the full pro of of Theorem 6 . In particular, Section 3.1 establishes the algebraic condition on K 1 , K 2 for exact KMS detailed balance and giv es their explicit construction. Section 3.2 app ends the rejection branch K and v eriﬁes the detailed balance and implemen tation properties. Finally , Section 3.3 reviews the single-tra jectory sampling proto col in [ JLL26 ], sets up the auto correla- tion time framework, and sho ws that the auto correlation time t aut con trols the decorrelation time b et ween consecutive samples, completing the pro of of Theorem 6 . 3.1 Construction of the Jump Branch W e begin by deﬁning the unnormalized jump branch of our measuremen t channel. Consider the completely p ositive map T ′ giv en b y tw o Kraus op erators K 1 , K 2 ∈ B ( H ): T ′ [ ρ ] := K 1 ρK † 1 + K 2 ρK † 2 . (17) W e require T ′ to satisfy the KMS detailed balance condition. The follo wing lemma establishes the algebraic condition on K 1 and K 2 that is suﬃcien t to guaran tee this. Lemma 7. If the Kr aus op er ators K 1 , K 2 ∈ B ( H ) satisfy K 2 = σ 1 / 2 β K † 1 σ − 1 / 2 β , (18) then the c ompletely p ositive map T ′ [ ρ ] in ( 17 ) satisﬁes KMS detaile d b alanc e with r esp e ct to σ β . Pr o of. By Deﬁnition 3 , T ′ is KMS detailed balanced if its Hilb ert–Schmidt adjoint T ′† [ X ] = K † 1 X K 1 + K † 2 X K 2 is self-adjoint under the KMS inner product. F or X , Y ∈ B ( H ), w e compute ⟨ X , T ′† ( Y ) ⟩ σ β , 1 / 2 = T r  X † σ 1 / 2 β K † 1 Y K 1 σ 1 / 2 β  + T r  X † σ 1 / 2 β K † 2 Y K 2 σ 1 / 2 β  . (19) F rom K 2 = σ 1 / 2 β K † 1 σ − 1 / 2 β , righ t-multiplying by σ 1 / 2 β giv es K 2 σ 1 / 2 β = σ 1 / 2 β K † 1 , and taking the adjoin t yields σ 1 / 2 β K † 2 = K 1 σ 1 / 2 β . Substituting in to ( 19 ) giv es T r  X † ( σ 1 / 2 β K † 1 ) Y ( K 1 σ 1 / 2 β )  = T r  X † ( K 2 σ 1 / 2 β ) Y ( σ 1 / 2 β K † 2 )  , 12 and T r  X † ( σ 1 / 2 β K † 2 ) Y ( K 2 σ 1 / 2 β )  = T r  X † ( K 1 σ 1 / 2 β ) Y ( σ 1 / 2 β K † 1 )  . Then, summing both iden tities implies, b y ( 19 ), ⟨ X , T ′† ( Y ) ⟩ σ β , 1 / 2 = ⟨ K † 2 X K 2 + K † 1 X K 1 , Y ⟩ σ β , 1 / 2 = ⟨T ′† ( X ) , Y ⟩ σ β , 1 / 2 , whic h establishes the KMS detailed balance of T ′ . W e now explicitly construct Kraus op erators K 1 and K 2 satisfying the algebraic condition ( 18 ) for KMS detailed balance and admitting eﬃcient blo ck enco dings for quantum implementation, using the ﬁltered observ able A f deﬁned in ( 12 ) and its imaginary-time shifted counterpart A ˜ f deﬁned in ( 15 ), resp ectively . Sp eciﬁcally , for a Hermitian normalized observ able A with || A || ≤ 1, w e emplo y the standard Gaussian ﬁlter f ( t ) = 1 √ 2 π τ 2 exp  − t 2 2 τ 2  with τ = Ω( β ) . W e deﬁne the ﬁrst Kraus op erator by K 1 := c p I + uA f , (20) where u ∈ (0 , 1) is the measurement strength and c > 0 is the scaling constant. Since f is real and ev en, A f is Hermitian. Moreov er, the Gaussian ﬁltered A f satisﬁes ∥ A f ∥ ≤ ∥ A ∥ ≤ 1, which gives I + uA f ≻ 0 for u ∈ (0 , 1), thus the matrix square ro ot in ( 20 ) is well-deﬁned, and K 1 is Hermitian and p ositive deﬁnite. T o satisfy the condition ( 18 ) for KMS detailed balance, we set K 2 := σ 1 / 2 β K 1 σ − 1 / 2 β = c σ 1 / 2 β p I + uA f σ − 1 / 2 β = c q I + uσ 1 / 2 β A f σ − 1 / 2 β . (21) By Lemma 5 , we ha ve σ 1 / 2 β A f σ − 1 / 2 β = A ˜ f with ˜ f ( t ) = f ( t − iβ / 2). Thus, ( 21 ) simpliﬁes to K 2 = c q I + uA ˜ f . (22) Since the shifted ﬁlter ˜ f tak es complex v alues, A ˜ f is not Hermitian. Nevertheless, Lemma 5 giv es ∥ A ˜ f ∥ ≤ exp  β 2 / (8 τ 2 )  ∥ A ∥ , which is O (1) when τ = Ω( β ). Moreo ver, for an y small measuremen t strength u ≤ 1 / (2 ∥ A ˜ f ∥ ), the op erator I + uA ˜ f has spectrum bounded a wa y from zero, and then the matrix square root in ( 22 ) is well-deﬁned. The quantum proto col and implemen tation cost of the blo ck enco dings of K 1 and K 2 are deferred to Section B . W e summarize the construction of K 1 and K 2 and their implemen tations in the following lemma. Lemma 8. L et U A b e a blo ck enc o ding of A . F or any tar get ac cur acy ϵ ′ > 0 , ϵ ′ -appr oximate blo ck en- c o dings of K 1 and K 2 c an b e c onstructe d using O (p olylog ( β ∥ H ∥ /ϵ ′ )) queries to U A , O ( β p olylog ( ∥ H ∥ /ϵ ′ )) c ontr ol le d H amiltonian simulation time, and O (p olylog ( β ∥ H ∥ /ϵ ′ )) ancil la qubits. The pro of follows b y combining Theorem 22 with Lemma 23 (for K 1 ) and Theorem 24 (for K 2 ). 13 Extracting Measuremen t Information from K 1 and K 2 . W e now sho w that observing either classical outcome asso ciated with K 1 or K 2 pro vides equiv alen t, un biased information ab out T r( σ β A ). First, the probability p 1 of obtaining the outcome corresponding to the K 1 -branc h with resp ect to the Gibbs state is p 1 = T r( σ β K † 1 K 1 ) = T r( σ β K 2 1 ) = c 2 T r( σ β ( I + uA f )) = c 2 (1 + u T r( σ β A )) , where we used T r( σ β A f ) = T r( σ β A ) from ( 13 ). This immediately yields an unbiased estimator for the thermal exp ectation T r( σ β A ). Similarly , using K 2 = σ 1 / 2 β K 1 σ − 1 / 2 β , the probability p 2 corresp onding to the K 2 -branc h is p 2 = T r( σ β K † 2 K 2 ) = T r  σ β ( σ − 1 / 2 β K 1 σ 1 / 2 β )( σ 1 / 2 β K 1 σ − 1 / 2 β )  = T r( σ β K 2 1 ) = p 1 . Th us, b oth branc hes yield the same outcome probabilit y , with the signal term u T r( σ β A ) scaling linearly with the measurement strength u . Assigning outcomes z = 1 to branc hes K 1 and K 2 , and z = 0 to the rejection branch K , the estimator v := z 2 c 2 u − 1 u satisﬁes E [ v ] = T r( σ β A ) in the ideal case, with v ariance of order ( cu ) − 2 . In our construction, u = O (1) and c = O (1 / log( β ∥ H ∥ )) (see Section 3.2 ), so the v ariance is of order O (log 2 ( β ∥ H ∥ )). 3.2 Construction of the Rejection Branch T o complete the measurement channel, we app end a rejection branch R [ ρ ] = K ρK † to comp ensate for the non-trace-preserving prop erty of T ′ , noting that T ′† [ I ] = K † 1 K 1 + K † 2 K 2 ≤ 3 c 2 I . (23) The construction of such a rejection branc h, whic h maintains the KMS detailed balance condition and eﬃcien t quantum implementabilit y , follows the discrete-time quantum Metrop olis-Hastings sampler framework of [ GCDK26 ]. Theorem 9. (T r ac e-Pr eserving Completion and Eﬃcient Implementation [ GCDK26 ]). Consider any σ β -detaile d b alanc e d c ompletely p ositive map T ′ such that its Heisenb er g-pictur e dual satisﬁes T ′† [ I ] ≤ I . The CP map, deﬁne d as M = T ′ + R , wher e R [ · ] = K [ · ] K † is exactly σ β -detaile d b alanc e d and tr ac e-pr eserving, given the r eje ction Kr aus op er ator: K := q √ σ β ( I − T ′† [ I ]) √ σ β σ − 1 / 2 β . In addition, for T ′ [ · ] = K 1 · K † 1 + K 2 · K † 2 with K 1 , K 2 deﬁne d in Se ction 3.1 with appr opriate p ar ameters c = O (1 / log( β ∥ H ∥ )) , u = O (1) , τ = O ( β + p olylog( β ∥ H ∥ /ϵ )) , the blo ck-enc o ding of K c an b e implemente d within ϵ ′ err or using polylog( β ∥ H ∥ /ϵ ′ ) queries to the blo ck enc o ding of T ′† [ I ] = K † 1 K 1 + K † 2 K 2 and β ∥ H ∥ p olylog ( β ∥ H ∥ /ϵ ′ ) queries to the blo ck enc o ding of H . Mor e over, the blo ck enc o ding of K † 1 K 1 + K † 2 K 2 c an b e implemente d using O (p olylog ( β ∥ H ∥ /ϵ ′ )) queries to the blo ck-enc o ding of A and O ( β p olylog ( β ∥ H ∥ /ϵ )) (c ontr ol le d)-Hamiltonian simulation time for H . 14 The k ey insigh t behind the eﬃcien t implemen tation is that [ GCDK26 ] sho ws the operator K admits a series expansion in which each term can b e written as an in tegral of short-time Hamiltonian ev olutions. The scaling c = O (1 / log ∥ H ∥ ) is chosen to mak e such series conv erge. When O is quasi-lo cal, and H is geometrically local, eac h term in this expansion preserv es locality , which in turn enables the eﬃcient quantum implemen tation. W e will state the mplementation result in App endix C ; w e refer the interested reader to [ GCDK26 ] for full details. R emark 1 . The implemen tation cost in Theorem 9 in volv es ˜ O ( β ∥ H ∥ ) queries to a blo ck enco ding of H /α . This matc hes the standard cost of blo c k-enco ding- and QSVT-based Hamiltonian simu- lation [ LC19 , GSL W19 ] for sim ulating H up to time ˜ O ( β ). W e therefore absorb this cost in to the Hamiltonian simulation cost rep orted in Theorem 6 . Ha ving established the blo ck enco dings of K 1 , K 2 , and K individually , we discuss in Ap- p endix B.3 how to com bine these comp onents to obtain an ϵ ′ -appro ximate implementation of the full measurement c hannel M . 3.3 The Single-T ra jectory Proto col With the detailed-balanced measuremen t c hannel M constructed in Sections 3.1 and 3.2 , we no w describ e how to com bine it with a Gibbs sampling channel N in the single-tra jectory framew ork of [ JLL26 ] to estimate T r( σ β A ). Proto col. Giv en a Gibbs sampling channel N that satisﬁes σ β -KMS detailed balance [ CK G25 , DLL25 ], the measuremen t c hannel M from Theorem 6 , an initial state ρ , and integers T burn and T , the proto col pro ceeds as follo ws: • Burn-in stage: Apply N for T burn steps to ρ to obtain a state close to σ β . • Sampling stage: F or eac h time step t = 1 , . . . , T , apply the comp osite sandwich c hannel E := M ◦ N ◦ M . • Measuremen t outcome: Let a t ∈ { 1 , 2 , 3 } denote the outcome branch lab el of the ﬁrst (righ tmost) application of M at step t . The real-v alued outcome is given b y the function v : { 1 , 2 , 3 } → R with v 1 = v 2 = 1 2 c 2 u and v 3 = 0. • Estimation: Output X T − 1 u , where X T := 1 T P T t =1 v a t , as the estimator of T r( σ β A ). Sample complexit y via the auto correlation time. Since b oth N and M satisfy exact KMS detailed balance with respect to σ β , the Gibbs state is the stationary state of the composite channel E . W e provide a self-contained treatmen t of the measurement outcome statistics and the relations b et ween the op erators deﬁned b elow in App endix A ; here w e outline the k ey quan tities. Let { K a } a =1 , 2 , 3 b e the Kraus op erators of M (with K 3 = K the rejection branch), and let v a ∈ R denote the real-v alued outcome asso ciated with branch a . Note that the statistics of M dep end on the choice of outcome function v ; here w e use v 1 = v 2 = 1 2 c 2 u and v 3 = 0 as deﬁned in the proto col. The mean and v ariance of the measuremen t outcome in the stationary state are: ¯ v = E σ β ( M ) := X a v a T r( K a σ β K † a ) , V ar σ β ( M ) := X a ( v a − ¯ v ) 2 T r( K a σ β K † a ) . 15 The statistical p erformance of the empirical av erage X T is gov erned b y the in tegrated auto correla- tion time of E : t aut ,T := 1 2 + T X t =1  1 − t T  Corr σ β ( E t − 1 ) V ar σ β ( M ) , (24) where the auto correlation function is C orr σ β ( E t ) := T r( b E v ◦ E t ◦ b E v ( σ β )) and b E v = P a ( v a − ¯ v ) E a . Here, c M ( X ) := P a ( v a − ¯ v ) K a X K † a is the cen tered measuremen t map. By [ JLL26 ] (see Lemma 16 for the precise statement), set T burn := t mix ( 1 3 η ), after the burn-in stage the current state ρ burn satisﬁes ∥ ρ burn − σ β ∥ ≤ 1 3 η . Then in the sampling stage, it suﬃces to set T = O  V ar σ β ( M ) ϵ 2 η t aut ,T  to estimate T r( σ β A ) to precision ϵ with failure probability 2 3 η for the exact implemen tation. Since V ar σ β ( M ) = O (log 2 ( β ∥ H ∥ )) (see Section 3.1 ), the total num b er of required steps is: T = O  log 2 ( β ∥ H ∥ ) ϵ 2 η t aut  . (25) Bounding the Auto correlation Time via the Sp ectral Gap. When the Gibbs sampler N has sp ectral gap λ > 0, the auto correlation time can b e b ounded b y O (1 /λ ). More sp eciﬁcally , b y [ JLL26 ] (see Prop osition 17 for the precise statemen t), if b oth N and M satisfy KMS detailed balance and the centered measuremen t map c M also satisﬁes KMS detailed balance, then: t aut ,T ≤ θ λ + 1 2 , where θ := P a,b ( v a − ¯ v )( v b − ¯ v )T r( K b K a σ β K † a K † b ) V ar σ β ( M ) . (26) W e now v erify that these conditions hold in our construction and that θ = O (1). Detaile d b alanc e of c M . Recall that c M ( X ) = P a ( v a − ¯ v ) K a X K † a . Since v 1 = v 2 = 1 2 c 2 u and v 3 = 0, the cen tered co eﬃcien ts are v 1 − ¯ v = v 2 − ¯ v = 1 2 c 2 u − ¯ v and v 3 − ¯ v = − ¯ v . Th us: c M ( X ) =  1 2 c 2 u − ¯ v   K 1 X K † 1 + K 2 X K † 2  − ¯ v K X K † =  1 2 c 2 u − ¯ v  T ′ [ X ] − ¯ v R [ X ] . Both T ′ and R individually satisfy KMS detailed balance: T ′ b y Lemma 7 , and R b y Theorem 9 . Since c M is a linear com bination of t wo KMS-detailed-balanced maps, c M also satisﬁes KMS detailed balance with respect to σ β . Bounding θ . The centered outcomes are v 1 − ¯ v = v 2 − ¯ v = 1 2 c 2 u − ¯ v = O (1 / ( c 2 u )) and v 3 − ¯ v = − ¯ v = O (1 /u ). Using T r( K b K a σ β K † a K † b ) ≤ ∥ K b ∥ 2 ∥ K a ∥ 2 and ∥ K 1 ∥ , ∥ K 2 ∥ = O ( c ), ∥ K 3 ∥ ≤ 1, the nominator in ( 26 ) is b ounded by O (1). Th us w e ha v e t aut ,T ≤ O (1) λ log 2 ( β ∥ H ∥ ) + 1 2 . Com bining this with ( 25 ), the total n um b er of required step is T = O  1 ϵ 2 η (log 2 ( β ∥ H ∥ ) + λ − 1 )  16 Stabilit y Analysis for the Appro ximate Implemen tation. In practice, the sandwich c hannel E = M ◦ N ◦ M is only implemented approximately . W e mo del this via the quan tum instrumen t framew ork of App endix A . Eac h step of the ideal proto col deﬁnes a quan tum instrumen t {E a } a =1 , 2 , 3 , where each branch E a ( ρ ) = M ◦ N ( K a ρK † a ) corresp onds to the Kraus op erator K a and carries the real-v alued outcome v a . A tra jectory of branc h lab els  a = ( a 1 , . . . , a T ) ∈ { 1 , 2 , 3 } T determines the sequence of real-v alued outcomes ( v a 1 , . . . , v a T ). The ideal tra jectory distribution ov er lab el sequences is induced by recursiv ely applying {E a } to the p ost-burn-in state: P (  a ) = T r( E a T ◦ · · · ◦ E a 1 ( ρ burn )) , where ρ burn . In the appro ximate protocol, w e instead apply an ϵ ′ -appro ximate instrumen t { ˜ E a } at eac h step (in the sense of Deﬁnition 13 ), inducing the approximate tra jectory distribution: ˜ P (  a ) = T r  ˜ E a T ◦ · · · ◦ ˜ E a 1 ( ρ burn )  . By Lemma 14 in App endix A , the total v ariation distance b etw een these tw o tra jectory distributions o ver T steps is b ounded b y: ∥ P − ˜ P ∥ TV ≤ 1 2 T ϵ ′ . W e now translate this TV b ound into a b ound on the failure probability of the estimator. The empirical av erage is X T = 1 T P T t =1 v a t , and the estimator is X T − 1 u . Deﬁne the failure even t S := {  a : | X T − 1 u − T r( σ β A ) | ≥ ϵ } . By the deﬁnition of total v ariation distance: ˜ P ( S ) ≤ P ( S ) + ∥ P − ˜ P ∥ TV . Since the exact proto col satisﬁes P ( S ) ≤ 2 3 η by Lemma 16 , w e obtain: ˜ P ( S ) ≤ 2 3 η + 1 2 T ϵ ′ . Setting ϵ ′ = O ( η /T ) mak es the second term at most η / 3, giving a total failure probability of η . Substituting T = O  log 2 ( β ∥ H ∥ ) ϵ 2 η t aut  , the required p er-step implemen tation accuracy of the measuremen t E is: ϵ ′ = O  ϵ 2 η 2 log 2 ( β ∥ H ∥ ) · t aut  . Since the implemen tation cost of M (Theorem 6 ) and N [ CK G25 , DLL25 ] scales logarithmically with the precision ϵ ′ , the cost incurred by the appro ximation introduces only a minor o verhead. 4 Measuremen t-Remixing Strategy: W arm-Start in χ 2 -Div ergence Our second construction is simpler than the detailed-balance approac h of Section 3 , at the cost of requiring a p ositive sp ectral gap λ > 0. W e design a tw o-outcome p ositiv e op erator-v alued mea- sure(PO VM) base d solely on the smo othed observ able A f with a real ﬁlter f , whose measuremen t outcomes yield an unbiased estimator of T r( σ β A ). Since f is real, the construction requires neither an imaginary-time shift nor a rejection branc h, making it more directly implementable than the detailed-balance channel. The k ey prop erty driving the analysis is that the post-selected state maintains a b ounded χ 2 - div ergence from σ β , ev en though the POVM does not satisfy detailed balance. Under a sp ectral 17 gap λ > 0, starting from this w arm start, the p ost-measurement state returns to within ϵ of σ β within time O ( λ − 1 log(1 /ϵ )), rather than the worst case mixing time. Since the remixing cost p er sample is en tirely indep endent of σ β , min , the O (log (1 /σ β , min )) factor in the mixing time appears only in the one-time burn-in, not in the p er-sample cost. W e assume ∥ A ∥ ≤ 1 without loss of generalit y . The main result is as follows. Theorem 10 (Measuremen t-Remixing Complexity) . F or any Hermitian observable A with ∥ A ∥ ≤ 1 , c onsider the r e al Gaussian ﬁlter f ( t ) = 1 √ 2 π τ 2 exp  − t 2 2 τ 2  . One c an cho ose τ = O ( β ) and the me asur ement str ength u = O (1) such that the me asur ement channel M [ ρ ] = K + ρK † + + K − ρK † − , K ± = r I ± uA f 2 , satisﬁes the fol lowing. Supp ose the Gibbs sampling channel N has sp e ctr al gap λ > 0 . Then: • Warm Start and R emixing Cost: L et ρ ′ ± denote the p ost-sele cte d state after applying M to ρ and r e c or ding outc ome ± , i.e., ρ ′ ± = K ± ρK ± / T r( ρK 2 ± ) . If χ 2 ( ρ, σ β ) = O (1) , then χ 2 ( ρ ′ ± , σ β ) = O (1) . Mor e over, k 0 = O  1 λ log 1 ϵ  steps of N suﬃc e to bring the p ost-sele cte d state b ack to ϵ -pr oximity of the Gibbs state: χ 2 ( N k 0 ( ρ ′ ± ) , σ β ) ≤ ϵ . • Eﬃcient Implementation: A n ϵ ′ -appr oximate implementation of M c an b e r e alize d with O (p olylog(1 /ϵ ′ )) queries to a blo ck enc o ding of A and O ( β p olylog (1 /ϵ ′ )) Hamiltonian simu- lation time for H , and O (log( β ∥ H ∥ ) + log log(1 /ϵ ′ )) ancil la qubits. • Sample Complexity: T o estimate T r( σ β A ) to ac cur acy ϵ with failur e pr ob ability η , start- ing fr om an initial state ρ burn satisfying ∥ ρ burn − σ β ∥ 1 ≤ η / 3 , the total numb er of r e quir e d me asur ement-r emixing steps is: T = O  log(1 /η ) ϵ 2  , pr ovide d e ach c ombine d me asur ement-r emixing step (applying M fol lowe d by N k 0 ) is imple- mente d to p er-step ac cur acy ϵ ′ = O  ϵ 2 η log(1 /η )  in the sense of Deﬁnition 13 . Compared to the detailed-balance approac h of Section 3 , this construction is more implementation- friendly: • F ewer ancilla qubits. On top of the ancillas needed for a blo ck enco ding of A f , imple- men ting K ± via QSVT requires only a constant n umber of additional ancilla qubits, since the target function p (1 ± ux ) / 2 is a real p olynomial (See [ GSL W19 ] and Section B.2.1 ). By con- trast, Section 3 requires p olylog ( β ∥ H ∥ /ϵ ) additional ancillas for the general (non-Hermitian) eigen v alue transformation implementing K 2 , plus further ancillas for the rejection op erator R (App endix C ). • No rejection branch. The rejection branch K in Section 3 is costly in tw o wa ys: it con tributes zero signal but increases the estimator v ariance b y a p olylog( β ∥ H ∥ ) factor, and its implementation via R requires an additional p olylog( β ∥ H ∥ ) ov erhead. Neither cost is presen t here. 18 • Better failure probabilit y dep endence. Because the measurement outcomes are b ounded, martingale concen tration yields sample complexit y T = O ( ϵ − 2 log(1 /η )), with only logarith- mic dependence on the failure probabilit y η . By con trast, the detailed-balance approac h relies on Chebyshev’s inequalit y , giving a 1 /η dep endence. W e note that a Mark ov c hain concen- tration inequalit y [ GGG23 ] could p oten tially impro ve the η − 1 dep endence in Theorem 6 to log η − 1 . In what follows, we detail the construction and analysis, building to w ard the full pro of of The- orem 10 . Section 4.1 details the explicit construction of the measuremen t c hannel M . Section 4.2 establishes the χ 2 -w arm-start guarantee (Theorem 11 ). Section 4.3 analyzes the sample complexity for estimating T r( σ β A ) from the measurement outcomes. 4.1 Construction of the Measuremen t Channel W e use the same real Gaussian ﬁlter f ( t ) = 1 √ 2 π τ 2 exp  − t 2 / 2 τ 2  as in Section 3 , so that A f is Hermitian with ∥ A f ∥ ≤ 1 and T r( σ β A f ) = T r( σ β A ). Fix a measuremen t strength u ∈ (0 , 1). The measuremen t c hannel is deﬁned by t wo Kraus op erators: K ± = q 1 2 ( I ± uA f ) . Since u ∥ A f ∥ ≤ u < 1, b oth 1 2 ( I ± uA f ) are positive semi-deﬁnite, and K † + K + + K † − K − = I , so M [ ρ ] = K + ρK † + + K − ρK † − is trace-preserving. The probability of outcome ± when the system is in state ρ is p ± ( ρ ) = T r  ρK † ± K ±  = 1 2  1 ± u T r( ρA f )  , so p + ( ρ ) − p − ( ρ ) = u T r( ρA f ). Assigning outcome + the v alue z = +1 and outcome − the v alue z = − 1, the estimator v := z u , where z ∈ { +1 , − 1 } is the observed outcome, satisﬁes E [ v ] = T r( ρA f ). The v ariance is (1 /u ) 2 = O (1) for u = O (1). The blo c k-enco ding implementation of K ± is discussed in Sections B.1 and B.2 , and the appro ximate implementation of the full POVM is discussed in Section B.3 . 4.2 The W arm-Start Prop ert y Up on observing outcome ± , the system collapses to the post-selected state ρ ′ ± = K ± ρK † ± p ± ( ρ ) , whic h is distinct from the p ost-measurement state M [ ρ ] a v eraged o ver outcomes. W e sho w that, for τ = Ω( β ), the p ost-selected state ρ ′ ± inherits the w arm-start prop ert y from ρ : if χ 2 ( ρ, σ β ) is b ounded, so is χ 2 ( ρ ′ ± , σ β ). Theorem 11. ( χ 2 -Warm-Start Pr op erty). L et χ 2 ( ρ, σ β ) ≤ C and deﬁne c f := exp  β 2 32 τ 2  . F or τ = Ω( β ) , c f = O (1) , and for me asur ement str ength u ∈ (0 , 1 /c f ) , the p ost-sele cte d state ρ ′ ± satisﬁes: χ 2 ( ρ ′ ± , σ β ) ≤ 4(1 + C ) (1 − u ) 2 (1 − c f u ) 2 . 19 F or u = O (1) and τ = Ω( β ), the righ t-hand side is O (1 + C ). In particular, when χ 2 ( ρ, σ β ) = O (1) after the burn-in p erio d, the p ost-selected state satisﬁes χ 2 ( ρ ′ ± , σ β ) = O (1) as w ell. This decouples the p er-sample remixing cost from σ β , min : the remixing duration to reac h ϵ ′ -pro ximity of σ β is t remix = O ( λ − 1 log(1 /ϵ ′ )), regardless of σ β , min . Pro of of Theorem 11 . W e ﬁrst establish that multiplication b y A f is b ounded in the ∥ · ∥ σ β , − 1 / 2 norm. By Lemma 5 . F or any op erator X : ∥ A f X ∥ σ β , − 1 / 2 ≤ ∥ σ 1 / 4 β A f σ − 1 / 4 β ∥ · ∥ X ∥ σ β , − 1 / 2 , and similarly for right m ultiplication. By Lemma 5 , the imaginary-time-shifted norm satisﬁes ∥ σ 1 / 4 β A f σ − 1 / 4 β ∥ = ∥ e − β H / 4 A f e β H / 4 ∥ ≤ ∥ A ∥ Z ∞ −∞ | f ( s − iβ / 4) | ds = ∥ A ∥ · c f , so ∥ A f X ∥ σ β , − 1 / 2 ≤ c f ∥ X ∥ σ β , − 1 / 2 and likewise ∥ X A f ∥ σ β , − 1 / 2 ≤ c f ∥ X ∥ σ β , − 1 / 2 . Let B ± = p I ± uA f , so that K ± = B ± / √ 2 and ρ ′ ± = B ± ρB ± / (2 p ± ( ρ )). Since p ± ( ρ ) = 1 2 (1 ± u T r( ρA f )) ≥ (1 − u ) / 2, we ha ve 2 p ± ( ρ ) ≥ 1 − u . F or c f u < 1, the generalized binomial series B ± = ∞ X m =0  1 / 2 m  ( ± uA f ) m con verges absolutely . Applying the m ultiplication b ound iteratively giv es ∥ A m f X A n f ∥ σ β , − 1 / 2 ≤ c m + n f ∥ X ∥ σ β , − 1 / 2 , so: ∥ B ± ρB ± ∥ σ β , − 1 / 2 ≤ ∞ X m =0      1 / 2 m      ( c f u ) m ! 2 ∥ ρ ∥ σ β , − 1 / 2 ≤ ∥ ρ ∥ σ β , − 1 / 2 1 − c f u , where the last inequality uses P m |  1 / 2 m  | x m ≤ (1 − x ) − 1 / 2 for x ∈ [0 , 1). Since p ± ( ρ ) ≥ (1 − u ) / 2, ∥ ρ ′ ± ∥ σ β , − 1 / 2 ≤ 2 ∥ ρ ∥ σ β , − 1 / 2 (1 − u )(1 − c f u ) . Since χ 2 ( ρ, σ β ) = ∥ ρ ∥ 2 σ β , − 1 / 2 − 1 ≤ C , w e hav e ∥ ρ ∥ σ β , − 1 / 2 ≤ √ 1 + C . Using χ 2 ( ρ ′ ± , σ β ) ≤ ∥ ρ ′ ± ∥ 2 σ β , − 1 / 2 completes the proof. 4.3 Ov erall Complexity With the measurement channel M and the warm-start guarantee of Theorem 11 in hand, we no w describ e the full estimation proto col and analyze its complexit y . Since M do es not satisfy detailed balance, each measuremen t is follow ed by a remixing phase of k 0 Gibbs sampling steps to return the state near σ β . W e analyze the sample complexity in the ideal case using martingale concen tration, exploiting the boundedness of the outcomes v a t to obtain log(1 /η ) dependence on the failure probabilit y . W e then p erform a stability analysis to account for appro ximate implementation errors. 20 Proto col. Giv en a Gibbs sampling c hannel N , the measurement channel M from Section 4.1 , an initial state ρ , and in tegers T burn , k 0 , and N , the proto col proceeds as follows: • Burn-in stage: Apply N for T burn := t mix ( η ) steps to obtain ρ burn satisfying ∥ ρ burn − σ β ∥ 1 ≤ η . Set ρ 1 = ρ burn . • Sampling stage: F or t = 1 , 2 , . . . , N : apply M to ρ t to obtain outcome label a t ∈ { + , −} and p ost-selected state ρ ′ t,a t = K a t ρK † a t T r( ρK † a t K a t ) ; then apply N k 0 to ρ ′ t,a t for k 0 = O  1 λ log 1 ϵ  to obtain state ρ t +1 = N k 0  ρ ′ t,a t  . • Estimation: Assign real-v alued outcomes v + = 1 /u and v − = − 1 /u to the branc h labels. Output 1 T P T t =1 v a t as the estimator of T r( σ β A ). Sample complexit y via mart ingale concen tration. W e ﬁrst analyze the ideal case where M and N are implemen ted exactly , and the initial state is exactly the Gibbs state, i.e. ρ 0 = σ β . Cho ose k 0 = O  1 λ log 1 ϵ  so that, b y Theorem 11 and the exponential decay of the χ 2 -div ergence under N , each remixed state s atisﬁes ∥ ρ t − σ β ∥ 1 ≤ ϵ 3 u for all t ≥ 1. Let F t denote the σ -algebra generated b y the outcome lab els ( a 1 , . . . , a t ) up to step t . The outcome v a t ∈ {− 1 /u, +1 /u } is F t -measurable with | v a t | ≤ 1 /u = O (1). Since ρ t is determined b y F t − 1 and ∥ ρ t − σ β ∥ 1 ≤ ϵ 3 u , the conditional bias satisﬁes: | E [ v a t | F t − 1 ] − T r( σ β A ) | ≤ 1 u ∥ ρ t − σ β ∥ 1 ≤ ϵ 3 . W e apply the following concen tration result: Lemma 12. L et ( X t ) N t =1 b e r e al-value d r andom variables adapte d to a ﬁltr ation ( F t ) , wher e F t is gener ate d by ( X 1 , . . . , X t ) . Supp ose | E [ X t | F t − 1 ] − µ | ≤ ϵ 3 and | X t − E [ X t | F t − 1 ] | ≤ c almost sur ely for al l t . Then for N ≥ 18 c 2 ϵ 2 log 6 η : P      1 N N X t =1 X t − µ      ≥ 2 ϵ 3 ! ≤ η 3 . Pr o of. Azuma’s inequalit y [ AS16 , Theorem 7.2.1]. Applying Lemma 12 to ( v a t ) with µ = T r( σ β A ) and c = 1 /u = O (1), w e conclude that T = O ( ϵ − 2 log(1 /η )) applications of the measurement channel M suﬃce to estimate T r( σ β A ) to accuracy 2 ϵ 3 with failure probability at most η 3 in the ideal case. The total num b er of Gibbs sampling steps is: T · k 0 = O  1 λ ϵ 2 log 1 η · log 1 ϵ  . Stabilit y analysis for appro ximate implemen tation. Each measurement step consists of applying M follo wed by N k 0 , whic h deﬁnes a quan tum instrumen t {E a } a ∈{ + , −} with E a ( ρ ) = N k 0 ( K a ρK † a ). A tra jectory of outcome lab els  a = ( a 1 , . . . , a T ) ∈ { + , −} T induces the ideal tra jec- tory distribution: P (  a ) = T r( E a T ◦ · · · ◦ E a 1 ( σ β )) . 21 In practice, M and N k 0 are only approximately implementable. W e instead apply an ϵ ′ -appro ximate instrumen t { ˜ E a } (in the sense of Deﬁnition 13 ) starting from ρ burn , inducing the approximate tra jectory distribution: ˜ P (  a ) = T r  ˜ E a T ◦ · · · ◦ ˜ E a 1 ( ρ burn )  . Since ∥ σ β − ρ burn ∥ 1 ≤ η 3 from the burn-in, Lemma 14 giv es: ∥ P − ˜ P ∥ TV ≤ 1 2  η 3 + T ϵ ′  . Deﬁne the failure ev ent S :=  | 1 T P T t =1 v a t − T r( σ β A ) | ≥ ϵ  . The ideal protocol satisﬁes P ( S ) ≤ η 3 b y the martingale analysis in Lemma 12 . Therefore: ˜ P ( S ) ≤ P ( S ) + ∥ P − ˜ P ∥ TV ≤ η 3 + η 6 + T ϵ ′ 2 . Setting T ϵ ′ 2 ≤ η 6 giv es total failure probability at most 2 η 3 ≤ η . Substituting T = O ( ϵ − 2 log(1 /η )), the required per-step accuracy is: ϵ ′ = O  ϵ 2 η log(1 /η )  . Since the implementation cost of M (Theorem 10 ) and N [ CK G25 , DLL25 ] scales logarith- mically with the precision ϵ ′ , the cost incurred b y the approximation introduces only a minor o verhead. Ac kno wledgmen ts B.L. is supported by National Key R&D Program of China, Gran t No. 2024YF A1016000. H.C. and L.Y. are supported by the U.S. Department of Energy , Oﬃce of Science, Accelerated Research in Quantum Computing Cen ters, Quantum Utilit y through Adv anced Computational Quan tum Algorithms, Grant No. DESC002557. J.J. is supp orted by the Simons Quantum P ostdo ctoral F ello wship and by a Simons In v estigator Aw ard in Mathematics through Gran t No. 825053. A The Measuremen t Pro cess and the Outcome Statistics T o provide a uniﬁed framework for analyzing the sample complexity (gov erned by the correla- tion time) and the stabilit y against n umerical implementation errors, we formalize our sequential measuremen t protocols using the language of quantum instrumen ts. W e describ e a general quan tum instrument, which yields a classical outcome branch lab el from a ﬁnite set a ∈ { 1 , 2 , . . . , s } , b y a collection of quan tum superop erators {E a } s a =1 that satisﬁes: • The sum ov er all p ossible measuremen t branc hes, denoted as E := P s a =1 E a , is a v alid quantum c hannel (i.e., a CPTP map) • Eac h individual map E a is completely positive (CP) and trace non-increasing. Eac h branch lab el a is asso ciated with a real-v alued outcome v a ∈ R via a function v : { 1 , . . . , s } → R . F or a quantum system initialized in a density matrix ρ , the application of this quantum instru- men t yields the branch lab el a with probability: p a = T r( E a ( ρ )) 22 Conditioned on observing branch a , the post-measurement state is updated to: ρ ( a ) = E a ( ρ ) T r( E a ( ρ )) = E a ( ρ ) p a In our framew ork for predicting observ ables, we apply this quan tum instrument sequen tially , whic h forms a measuremen t pro cess. Starting from an initial state ρ 0 , we apply the instrumen t at discrete time steps t = 1 , 2 , . . . , T . Let  a = ( a 1 , a 2 , . . . , a T ) ∈ { 1 , . . . , s } T denote a speciﬁc sequence, or tra jectory , of recorded branc h lab els o v er these T steps, with corresp onding real-v alued outcomes v a t . The joint probabilit y distribution of observing a sp eciﬁc tra jectory  a is deﬁned by: P (  a ) = T r  E a T ◦ E a T − 1 ◦ · · · ◦ E a 1 ( ρ 0 )  W e can no w cleanly map this uniﬁed framework onto the tw o primary measurement strategies dev elop ed in this paper. Detailed-Balanced Measuremen t (Section 3 ) F or the detailed-balanced construction, the measuremen t part is decomp osed in to three branc hes: M = M 1 + M 2 + M 3 , where eac h branc h acts via a single Kraus op erator M a ( ρ ) = K a ρK † a , where K a is deﬁned in Theorem 6 , with real- v alued outcomes v 1 = v 2 = 1 and v 3 = 0. Let N denote the one-step Gibbs sampling channel. The eﬀectiv e single-step map of the quantum instrumen t corresponding to observing branc h lab el a is: E a = M ◦ N ◦ M a Summing ov er all branc hes yields the aggregate unconditional channel E = M ◦ N ◦ M . Measuremen t-Remixing (Section 4 ) F or the warm-start strategy , the measurement c hannel consists of t wo branc hes of measurement outcomes: M = M + + M − , where M ± ( ρ ) = K ± ρK † ± is deﬁned in Section 4.1 . Let N denote the single-step Gibbs sampling c hannel and k 0 denote a time that suﬃces for the remixing from warm start. In this case, the eﬀective map for outcome i is E ± = N k 0 ◦ M ± . A.1 Stabilit y Analysis of Implementation Errors In practice, executing these op e rations on a quantum computer in tro duces implementation errors (e.g., from blo ck-encoding). Instead of the exact measuremen t branc hes M a and the ideal mixing c hannel N , we implemen t approximations. W e quan tify this n umerical accuracy using the follo wing deﬁnition: Deﬁnition 13. W e sa y that a quan tum instrumen t { ˜ E a } s a =1 is an ϵ -appro ximate implemen tation of the ideal quantum instrument {E a } s a =1 if, for an y input densit y matrix ρ , the single-step statistical deviation is bounded b y: s X a =1 ∥ ˜ E a ( ρ ) − E a ( ρ ) ∥ 1 ≤ ϵ. 23 This error mo del translates in to a sequence of noisy tra jectory maps. Let ˜ P (  a ) denote the probabilit y distribution of the tra jectory branch lab els generated by recursiv ely applying the ap- pro ximate instrument { ˜ E a } s a =1 to the initial state ρ 0 . By b ounding the distance b et ween the exact comp osition and its noisy counterpart, we can rigorously b ound the total v ariation distance betw een the ideal and physically realized tra jectory distributions. Lemma 14 (Error Accum ulation in Sequential Measuremen ts) . L et P b e the tr aje ctory distribution gener ate d by the ide al quantum instrument {E a } s a =1 over T steps starting fr om ρ 0 , and ˜ P b e the distribution gener ate d by an ϵ -appr oximate implementation { ˜ E a } s a =1 starting fr om ˜ ρ 0 , with ∥ ρ 0 − ˜ ρ 0 ∥ 1 ≤ η . The T otal V ariation (TV) distanc e b etwe en the two tr aje ctory distributions is b ounde d by: ∥ P − ˜ P ∥ TV ≤ 1 2 ( η + T ϵ ) . Pr o of. T o b ound the TV distance b etw een the classical tra jectory distributions, w e shall embed the classical outcomes in to auxiliary quantum registers. T o do so, let H S denote the Hilb ert space of the quan tum system. At eac h time step t ∈ { 1 , . . . , T } , we introduce a fresh classical register C t . Let C 0 , the c ovarianc e b etwe en outc omes at steps t and t + p in the stationary state is: E σ β  ( v a t − ¯ v )( v a t + p − ¯ v )  = C orr σ β ( E p − 1 ) . Pr o of. The join t probability of observing a t = a and a t + p = b , marginalizing o ver the p − 1 in termediate steps, is T r( E b ◦ E p − 1 ◦ E a ( σ β )) (using stationarit y). Therefore: E σ β  ( v a t − ¯ v )( v a t + p − ¯ v )  = X a,b ( v a − ¯ v )( v b − ¯ v ) T r  E b ◦ E p − 1 ◦ E a ( σ β )  = T r  b E v ◦ E p − 1 ◦ b E v ( σ β )  , whic h equals C orr σ β ( E p − 1 ) by deﬁnition. The integrated (ﬁnite-size) auto correlation time for a tra jectory of length T is: t aut ,T := 1 2 + T X t =1  1 − t T  C orr σ β ( E t − 1 ) V ar σ β ( E ) . (30) Using Lemma 15 and the deﬁnition ( 30 ), one veriﬁes directly that the v ariance of the empirical mean satisﬁes: V ar σ β 1 T T X t =1 v a t ! = 1 T 2 T X t,s =1 E σ β [( v a t − ¯ v )( v a s − ¯ v )] = 1 T 2   T V ar σ β ( E ) + 2 T − 1 X p =1 ( T − p ) C orr σ β ( E p − 1 )   = 2 t aut ,T V ar σ β ( E ) T , (31) where the second equality uses translation in v ariance of the pro cess in the stationary state together with Lemma 15 . Lemma 16 ([ JLL26 ]) . L et {E a } b e a quantum instrument satisfying KMS detaile d b alanc e with the stationary state σ β . L et ρ b e an initial state with ∥ ρ − σ β ∥ 1 ≤ η . Deﬁne the tr aje ctory distributions P ρ (  a ) := T r( E a T ◦ · · · ◦ E a 1 ( ρ )) , P σ β (  a ) := T r( E a T ◦ · · · ◦ E a 1 ( σ β )) . F or any ϵ > 0 , if T ≥ 2 V ar σ β ( E ) ϵ 2 η t aut ,T , then P ρ      1 T T X t =1 v a t − E σ β ( E )      ≥ ϵ ! ≤ 2 η . 26 Pr o of. Let S := {  a : | 1 T P t v a t − E σ β ( E ) | ≥ ϵ } b e the failure even t. Applying Lemma 14 with b oth the ideal and the comparison pro cess using the same instrument {E a } (so the p er-step approximation error is ϵ ′ = 0), starting from ρ 0 = ρ and ˜ ρ 0 = σ β resp ectiv ely , with ∥ ρ 0 − ˜ ρ 0 ∥ 1 = ∥ ρ − σ β ∥ 1 ≤ η , w e obtain: ∥ P ρ − P σ β ∥ TV ≤ 1 2 ( η + T · 0) = η 2 . By the Cheb yshev inequality and ( 31 ), P σ β ( S ) ≤ η . Therefore: P ρ ( S ) ≤ P σ β ( S ) + ∥ P ρ − P σ β ∥ TV ≤ η + η 2 ≤ 2 η . F or the sp eciﬁc structure E a = M ◦ N ◦ M a arising in b oth of our measurement strategies, the cen tered instrument map takes the form b E v = M ◦ N ◦ c M , where c M := P a ( v a − v ) M a is the cen tered measuremen t map. In this case, the auto correlation time admits a spectral bound: Prop osition 17 ([ JLL26 ]) . Supp ose E a = M ◦ N ◦ M a , wher e N and M satisfy KMS detaile d b alanc e with r esp e ct to σ β , and the c enter e d me asur ement map c M = P a ( v a − v ) M a also satisﬁes KMS detaile d b alanc e. If N has sp e ctr al gap λ > 0 , then: t aut ,T ≤ θ λ + 1 2 , wher e θ := P a,b ( v a − v )( v b − v ) T r( M b ◦ M a ( σ β )) V ar σ β ( E ) . B Implemen tation of the Measurement Channels F or completeness, w e review the deﬁnition of blo ck enco dings of non-unitary matrices. Deﬁnition 18. A unitary U A is said to b e an ( α , b, ϵ )-block enco ding of a matrix A if the top-left blo c k of U A appro ximates A/α as follo ws: ∥ A − α ( ⟨ 0 b | ⊗ I ) U A ( | 0 b ⟩ ⊗ I ) ∥ ≤ ϵ, where α is called the sub-normalization factor. W e collect the k ey arithmetic properties of block enco dings used throughout the pap er [ LC19 , BCC + 14 , CLBY23 , CGJ19 , Lin22 ], which allo w us to construct complex blo ck enco dings from simpler ones. Lemma 19. • (Pr o duct of Blo ck Enc o dings) L et A 0 , . . . , A M − 1 b e matric es with c omp atible dimensions. Assume that for e ach j ∈ { 0 , 1 , . . . , M − 1 } , we ar e given an ( α j , b j , ϵ j ) -blo ck enc o ding U A j . L et U b e the unitary obtaine d by applying U A 0 , U A 1 , . . . , U A M − 1 se quential ly on disjoint ancil la r e gisters and a c ommon system r e gister. Then U is a blo ck enc o ding of A M − 1 · · · A 1 A 0 with p ar ameters   M − 1 Y j =0 α j , M − 1 X j =0 b j , M − 1 Y j =0 ( α j + ϵ j ) − M − 1 Y j =0 α j   . In p articular, if U A is an ( α, b, ϵ ) -blo ck enc o ding of A and M ϵ ≤ α , then applying M c opies of U A on disjoint ancil la r e gisters yields an ( α M , M b, O ( M α M − 1 ϵ )) -blo ck enc o ding of A M . 27 • (Line ar Combination of Blo ck Enc o dings) L et A 0 , . . . , A M − 1 b e matric es and let c 0 , . . . , c M − 1 ∈ C , wher e c j = | c j | e iϕ j . F or e ach j ∈ { 0 , . . . , M − 1 } , supp ose U A j is an ( α j , b j , ϵ j ) -blo ck en- c o ding of A j . L et b := max j b j and r e gar d e ach U A j as acting on a c ommon b -qubit ancil la r e gister by p adding with unuse d ancil las when ne c essary. L et m := ⌈ log 2 M ⌉ , and deﬁne the pr ep ar e or acle U prep on the m -qubit c ontr ol r e gister by U prep | 0 m ⟩ = 1 √ γ M − 1 X j =0 q | c j | α j | j ⟩ , γ := M − 1 X j =0 | c j | α j . The sele ct or acle for ac c essing U A j is deﬁne d by U sel = M − 1 X j =0 | j ⟩⟨ j | ⊗ e iϕ j U A j + 2 m − 1 X j = M | j ⟩⟨ j | ⊗ I . Then, the matrix W := ( U † prep ⊗ I ) U sel ( U prep ⊗ I ) is a ( γ , m + b, P M − 1 j =0 | c j | ϵ j ) -blo ck enc o ding of P M − 1 j =0 c j A j . B.1 Blo c k enco ding of A f and A ˜ f T o implemen t the detailed-balanced measurement c hannel of Section 3 and the POVM scheme of Section 4 , w e require eﬃcien t block enco dings of the ﬁltered observ ables A g := Z R g ( t ) e iH t Ae − iH t dt, g ∈ { f , ˜ f } . (32) Here f ( t ) = 1 √ 2 π τ exp  − t 2 2 τ 2  , ˜ f ( t ) = f  t − iβ 2  . (33) The op erator A f is Hermitian, while A ˜ f is generally non-Hermitian. Both app ear in the construc- tions of Section 3 , and A f is also the ﬁltered observ able used in the w arm-start PO VM of Section 4 . W e ﬁrst state a quadrature lemma for op erator-v alued in tegrals, which essentially follows from the Nyquist–Shannon sampling theorem. Lemma 20 (Quadrature lemma) . L et ϕ : R → B ( H ) b e inte gr able and admit a F ourier tr ansform b ϕ ( ω ) = Z R e − iω t ϕ ( t ) d t. F or any even inte ger M > 0 and step size ∆ t > 0 , we have       ∆ t M / 2 − 1 X k = − M / 2 ϕ ( k ∆ t ) − Z R ϕ ( t ) d t       ≤ X n  =0     b ϕ  2 π n ∆ t      | {z } aliasing err or + ∆ t X | k |≥ M / 2 ∥ ϕ ( k ∆ t ) ∥ | {z } trunc ation err or . Pr o of. By the Poisson summation formula [ Pin02 , Theorem 4.2.2] applied to the discrete grid: ∆ t X k ∈ Z ϕ ( k ∆ t ) = X n ∈ Z b ϕ  2 π n ∆ t  , 28 and noting b ϕ (0) = Z ∞ −∞ e − i (0) t ϕ ( t ) dt = Z ∞ −∞ ϕ ( t ) dt, w e ha ve ∆ t M / 2 − 1 X k = − M / 2 ϕ ( k ∆ t ) − Z ∞ −∞ ϕ ( t ) dt = X n  =0 b ϕ  2 π n ∆ t  − ∆ t X k / ∈ [ − M / 2 ,M / 2 − 1] ϕ ( k ∆ t ) , where the ﬁrst term on the right is the aliasing error from the ﬁnite grid spacing (the non-zero F ourier modes), and the second term is the truncation error from cutting oﬀ the inﬁnite sum at M / 2. The pro of is complete b y the triangle inequality . W e next collect the elementary estimates for the tw o ﬁlters f and ˜ f . Lemma 21 (Gaussian ﬁlter estimates) . F or the two ﬁlter functions g ∈ { f , ˜ f } deﬁ ne d ab ove, it holds that 1. Time-domain de c ay: | g ( t ) | ≤ C g √ 2 π τ e − t 2 / (2 τ 2 ) , C f = 1 , C ˜ f = e β 2 / (8 τ 2 ) . (34) 2. F ourier tr ansforms: b f ( ω ) = e − τ 2 ω 2 / 2 , b ˜ f ( ω ) = e β 2 / (8 τ 2 ) exp  − τ 2 2  ω − β 2 τ 2  2  . (35) 3. L 1 norms: ∥ f ∥ L 1 = 1 , ∥ ˜ f ∥ L 1 = e β 2 / (8 τ 2 ) . (36) In p articular, if τ = Ω( β ) , then ∥ g ∥ L 1 = O (1) for b oth g = f and g = ˜ f . W e now deriv e the resulting block-encoding costs of A f and A ˜ f . Theorem 22 (Blo ck enco ding of the ﬁltered observ ables) . L et H b e a system Hamiltonian, and A b e a Hermitian observable with ∥ A ∥ ≤ 1 . Supp ose that U A is an ( α A , b A , ϵ A ) -blo ck enc o ding of A . L et g ∈ { f , ˜ f } , wher e f and ˜ f ar e given in ( 33 ) with τ = Ω( β ) . Then, for any ϵ > 0 , one c an c onstruct an ( α g , b g , ϵ g ) -blo ck enc o ding of A g = Z R g ( t ) e iH t Ae − iH t d t , with the normalization factor α g , the numb er of ancil las b g : α g = O ( α A ) , b g = b A + O  log( τ ∥ H ∥ ) + log log (1 /ϵ )  , and the ac cur acy ϵ g = O ( ϵ A + ϵ ) . Mor e over, the implementation uses one query to U A and the c ontr ol le d H amiltonian simulation time T = O  τ p log(1 /ϵ )  . 29 Pr o of of The or em 22 . W e divide the pro of into four steps. Step 1: trunc ation of the inte gr al. Deﬁned the truncated integral: A ( T ) g := Z T − T g ( t ) e iH t Ae − iH t d t . Since unitary conjugation preserves the op erator norm and ∥ A ∥ ≤ 1, ∥ A g − A ( T ) g ∥ ≤ Z | t | >T | g ( t ) | dt. F or g = f and g = ˜ f , we ha ve, by ( 34 ) in Lemma 21 , ∥ A g − A ( T ) g ∥ ≤ 2 C g √ 2 π τ Z ∞ T e − t 2 / (2 τ 2 ) dt = O  C g e − T 2 / (2 τ 2 )  . Therefore, choosing T = O  τ q log( C g /ϵ )  (37) mak es the truncation error at most ϵ/ 3. In the regime τ = Ω( β ), we ha ve C g = O (1) for b oth g = f and g = ˜ f , and then T = O  τ p log(1 /ϵ )  . Step 2: discr etization by a uniform grid. Let t j := − T + j ∆ t, j = 0 , . . . , M − 1 , ∆ t := 2 T M , and deﬁne the discretized op erator e A g := ∆ t M − 1 X j =0 g ( t j ) e iH t j Ae − iH t j . (38) F or conv enience, we assume M = 2 m for an integer m > 0. Since T = M ∆ t/ 2, the grid { t j } M − 1 j =0 coincides with { k ∆ t } M / 2 − 1 k = − M / 2 in Lemma 20 up on the relab eling k = j − M / 2. W e apply the quadrature b ound of Lemma 20 to the op erator-v alued function ϕ ( t ) := g ( t ) e iH t Ae − iH t . Its F ourier transform is b ϕ ( ω ) = X k,ℓ b g  ω − ( E k − E ℓ )  Π k A Π ℓ , where H = P k E k Π k is the spectral decomp osition of H . Since | E k − E ℓ | ≤ 2 ∥ H ∥ and ∥ A ∥ ≤ 1, w e obtain ∥ b ϕ ( ω ) ∥ ≤ sup | x |≤ 2 ∥ H ∥ | b g ( ω − x ) | . 30 F or the t w o ﬁlters under consideration, w e ha ve the F ourier transforms ( 35 ), and th us in b oth cases, there holds | b g ( ω ) | ≤ C g exp  − τ 2 2 ( ω − µ g ) 2  , with µ f = 0 , µ ˜ f = β 2 τ 2 . Therefore, we ﬁnd ∥ b ϕ ( ω ) ∥ ≤ C g sup | x |≤ 2 ∥ H ∥ exp  − τ 2 2 ( ω − x − µ g ) 2  . Applying Lemma 20 , the aliasing term is bounded b y X n  =0     b ϕ  2 π n ∆ t      ≤ 2 C g X n ≥ 1 exp  − τ 2 2  2 π n ∆ t − 2 ∥ H ∥ − µ g  2  , (39) for ∆ t − 1 = Ω( ∥ H ∥ + µ g ). Let a := 2 π ∆ t , b := 2 ∥ H ∥ + µ g . If a > b , then for all n ≥ 1, there holds an − b ≥ n ( a − b ). Therefore, ( 39 ) giv es X n  =0     b ϕ  2 π n ∆ t      ≤ 2 C g X n ≥ 1 exp  − τ 2 2 n 2 ( a − b ) 2  ≤ 2 C g exp h − τ 2 2 ( a − b ) 2 i 1 − exp h − τ 2 2 ( a − b ) 2 i . Then, choosing a − b = Θ  τ − 1 q log( C g /ϵ )  , equiv alen tly , ∆ t − 1 = Θ  ∥ H ∥ + µ g + τ − 1 q log( C g /ϵ )  , mak es the aliasing error at most ϵ/ 3. Moreov er, we can see that the truncation error term in Lemma 20 is b ounded b y ∆ t X | j |≥ M / 2 ∥ ϕ ( j ∆ t ) ∥ ≤ ∆ t X | j |≥ M / 2 | g ( j ∆ t ) | = O ( ϵ ) , if T = O  τ q log( C g /ϵ )  , whic h is already ensured b y Step 1. Th us, w e hav e prov ed ∥ e A g − A g ∥ ≤ ϵ. F urther, a direct computation gives M = 2 T ∆ t = O  τ ∥ H ∥ q log( C g /ϵ ) + log ( C g /ϵ ) + τ µ g q log( C g /ϵ )  . 31 F or g = f , w e hav e µ g = 0, while for g = ˜ f , if τ = Ω( β ) then τ µ g = β / (2 τ ) = O (1). Therefore, for b oth cases, w e ha v e M = O  τ ∥ H ∥ p log(1 /ϵ ) + log(1 /ϵ )  . Step 3: LCU blo ck enc o ding of the discr etize d op er ator. F or each grid p oin t t j , deﬁne W j := ( I anc ⊗ e iH t j ) U A ( I anc ⊗ e − iH t j ) . (40) Since conjugation by unitaries preserves blo c k-enco ding parameters, each W j is an ( α A , b A , ϵ A )- blo c k enco ding of e iH t j Ae − iH t j . W e write c j := ∆ t g ( t j ) and recall e A g in ( 38 ). By linear combina- tions of unitaries (Lemma 19 ), there is a blo ck enco ding of e A g with normalization e α g := α A M − 1 X j =0 | c j | = α A ∆ t M − 1 X j =0 | g ( t j ) | , ancilla cost b A + log 2 M , and implemen tation error e ϵ g := ϵ A ∆ t P M − 1 j =0 | g ( t j ) | . It remains to estimate e α g and e ϵ g . It suﬃces to consider the quadrature estimate of | g ( t ) | . Using the same c hoices of T and ∆ t as abov e, we can derive ∆ t M − 1 X j =0 | g ( t j ) | = ∥ g ∥ L 1 + O ( ϵ ) , whic h readily gives, b y ( 36 ), e α g = α A ( ∥ g ∥ L 1 + O ( ϵ )) = α A ( C g + O ( ϵ )) , e ϵ g = ( C g + O ( ϵ )) ϵ A . Com bining this with the discretization error ∥ e A g − A g ∥ ≤ ϵ yields an ( α g , b g , ϵ g )-blo c k encoding of A g with desired parameters. Step 4: query c omplexity and c ontr ol le d evolutions. The select oracle tak es the form U sel = M − 1 X j =0 | j ⟩⟨ j | ⊗ e iϕ j W j , c j = | c j | e iϕ j = | ∆ t g ( t j ) | e iϕ j . Substituting the deﬁnition ( 40 ) of W j , we obtain the factorization U sel =  X j | j ⟩⟨ j | ⊗ I anc ⊗ e iϕ j e iH t j  ( I ⊗ U A )  X j | j ⟩⟨ j | ⊗ I anc ⊗ e − iH t j  . Therefore, the whole construction uses exactly one query to U A . The required con trolled Hamilto- nian evolutions only inv olve times t j ∈ [ − T , T ], and hence the maxim um ev olution time is of the same order as the truncation time T in ( 37 ). This prov es the theorem. B.2 Blo c k enco ding of c p I + uA f and c p I + uA ˜ f Ha ving obtained an eﬃcient blo ck enco ding of the ﬁltered op erator A g (whic h represen ts either A f or A ˜ f ) in Theorem 22 , our next task is to build blo c k encodings of the square-root operators c p I + uA g , g ∈ { f , ˜ f } , 32 from a block enco ding of A g . W e record the T aylor series of the square ro ot function for later use. The binomial series gives, for complex z ∈ C with | z | < 1, √ 1 + z = ∞ X k =0  1 / 2 k  z k , (41) where the co e ﬃcien ts satisfy    1 / 2 k    ≤ 1 for all k ≥ 0. The absolute con vergence of the series ( 41 ) is guaranteed b y ∞ X k =0      1 / 2 k      r k = 2 − √ 1 − r , r ∈ [0 , 1) . (42) T runcating ( 41 ) at order K yields the polynomial P K ( z ) := K X k =0  1 / 2 k  z k , with error   √ 1 + z − P K ( z )   ≤ ∞ X k = K +1 | z | k = | z | K +1 1 − | z | , | z | < 1 . (43) Hence, for | z | ≤ r < 1, choosing K = O  log(1 /ϵ ) log(1 /r )  ensures the truncation error is O ( ϵ ). B.2.1 The QSVT construction for real ﬁlters When g = f is real, the ﬁltered observ able A f is Hermitian, so one can apply QSVT [ GSL W19 ] to implemen t c p I + uA f . In particular, given a block encoding of A f , this requires only a constant n umber of additional ancilla qubits. Lemma 23. Supp ose U B is an ( α, b, ϵ 0 ) -blo ck enc o ding of a Hermitian matrix B . L et r := | u | α ≤ 1 2 , c ∈ (0 , 1 4 ] . (44) Then, for any given ϵ ′ ∈ (0 , 1 8 ] , ther e exists a (1 , b + 2 , ϵ ) -blo ck enc o ding of c √ I + uB with ϵ = 4 d p ϵ 0 /α + 2 ϵ ′ , using d = O (log(1 /ϵ ′ )) queries to U B and U † B , one query to c ontr ol le d- U B , and O (( b +1) d ) additional one- and two-qubit gates. Pr o of. W e apply Theorem 56 and Corollary 66 of [ GSL W19 ]. By deﬁnition, U B enco des e B := B /α , whose eigenv alues lie in [ − 1 , 1]. W e apply the function f ( x ) = c √ 1 + uαx to e B , so that f ( e B ) = c √ I + uB . 33 W e construct a p olynomial appro ximation of f on [ − 1 , 1] via [ GSL W19 , Corollary 66], whic h states: if f ( x ) = P ∞ ℓ =0 a ℓ x ℓ con verges for | x | ≤ 1 + δ and P ∞ ℓ =0 | a ℓ | (1 + δ ) ℓ ≤ M , then f can be ϵ ′ -appro ximated on [ − 1 , 1] by a p olynomial P d of degree d = O  δ − 1 log  M /ϵ ′  with ∥ P d ∥ L ∞ ([ − 1 , 1]) ≤ M . The T a ylor co eﬃcients of f are a ℓ = c  1 / 2 ℓ  ( uα ) ℓ , and the radius of conv ergence is 1 /r > 2 since r = | u | α ≤ 1 / 2 b y ( 44 ). Hence, w e can tak e δ = 1. Using ( 42 ), we ﬁnd M = ∞ X ℓ =0 | a ℓ | 2 ℓ = c ∞ X ℓ =0      1 / 2 ℓ      (2 r ) ℓ = c  2 − √ 1 − 2 r  ≤ 2 c ≤ 1 2 . Therefore, there exists a real p olynomial P d of degree d = O (log (1 /ϵ ′ )) such that ∥ f − P d ∥ L ∞ ([ − 1 , 1]) ≤ ϵ ′ , ∥ P d ∥ L ∞ ([ − 1 , 1]) ≤ 1 2 . By [ GSL W19 , Theorem 56], applying QSVT to the ( α, b, ϵ 0 )-blo c k encoding U B yields a (1 , b + 2 , ϵ ′′ )-blo c k enco ding of P d ( e B ) with ϵ ′′ = 4 d p ϵ 0 /α + ϵ ′ , using d queries to U B and U † B , one controlled- U B , and O (( b + 1) d ) additional gates. Finally , since ∥ f − P d ∥ L ∞ ([ − 1 , 1]) ≤ ϵ ′ , the triangle inequality giv es ϵ = ϵ ′′ + ϵ ′ = 4 d p ϵ 0 /α + 2 ϵ ′ . B.2.2 The LCU construction for general ﬁlters When the ﬁltered op erator is non-Hermitian (for example, A ˜ f ), QSVT is not directly applicable for constructing a blo c k encoding of the asso ciated matrix function. Instead, w e use a T a ylor-series expansion together with LCU to implemen t the block enco ding of √ I + uB for a general operator B . Alternative quantum eigen v alue-transformation metho ds may also b e applicable in this setting; see, for example, [ TOSU20 , LS24 , A CL Y24 ]. Theorem 24. L et B b e an op er ator with an ( α, b, δ ) -blo ck enc o ding U B . L et u, c ∈ C satisfy r := | u | α ≤ 1 2 , | u | ( α + δ ) < 1 . (45) Then, for any given ϵ > 0 , one c an c onstruct a ( γ , b ′ , ϵ ′ ) -blo ck enc o ding of c √ I + uB with γ = c  2 − √ 1 − r  = O ( c ) , b ′ = O  b log(1 /ϵ ) log(1 /r )  , and ϵ ′ = ϵ + O ( δ ) . (46) using K = O  log(1 /ϵ ) log(1 /r )  queries to U B . 34 Pr o of. W e expand c √ I + uB = c P ∞ k =0  1 / 2 k  u k B k and truncate at order K , deﬁning P K := c K X k =0  1 / 2 k  u k B k . By ( 43 ) and | u |∥ B ∥ ≤ | u | α = r , we hav e ∥ c √ I + uB − P K ∥ ≤ c r K +1 1 − r . Th us the choice of K = O  log(1 /ϵ ) log(1 /r )  ensures the truncation error is at most ϵ . By Lemma 19 (pro duct of blo ck enco dings), the k -th p ow er B k admits an ( α k , kb, ( α + δ ) k − α k ) blo c k enco ding. Expressing P K as an LCU and applying Lemma 19 again yields the normalization constan t γ = c K X k =0      1 / 2 k      ( | u | α ) k ≤ c ∞ X k =0      1 / 2 k      r k = c  2 − √ 1 − r  , the ancilla coun t b ′ = K b + ⌈ log 2 ( K + 1) ⌉ = O  b log(1 /ϵ ) log(1 /r )  , and the block enco ding error, b y | u | ( α + δ ) < 1 and ( 42 ), c K X k =0      1 / 2 k      | u | k  ( α + δ ) k − α k  = c   2 − p 1 − r − | u | δ  −  2 − √ 1 − r   = c  √ 1 − r − p 1 − r − | u | δ  = O  | u | δ √ 1 − r  . Since r ≤ 1 / 2 and | u | δ < 1 − r by ( 45 ), w e conclude with O ( δ ). Finally , the select oracle U sel = P K k =0 | k ⟩⟨ k | ⊗ U k B is implemented using K independent b -qubit ancilla blo cks: for each i = 1 , . . . , K , apply U B to the system and the i -th ancilla blo ck, controlled on the predicate k ≥ i . This uses exactly K queries to U B . Com bining the truncation and enco ding errors completes the pro of of ( 46 ). B.3 Appro ximate Implementation of the PO VM Theorem 25 (Appro ximate implemen tation of quan tum instruments) . L et s = O (1) , c onsider a quantum instrument {M i } s − 1 i =0 given by the Kruas op er ators: M i ( ρ ) = K i ρK † i , s − 1 X i =0 K † i K i = I . 35 Supp ose that for e ach i , the ( α i , b i , ϵ ) -blo ck enc o ding of K i ar e available, denote d as U i . L et α = max i α i and b = max i b i . Then ther e exists a quantum cir cuit that implements an appr oximation { f M i } s − 1 i =0 such that, for any density matrix ρ , s − 1 X i =0 ∥ f M i ( ρ ) − M i ( ρ ) ∥ 1 = O ( ϵ ) , using O ( α log(1 /ϵ )) queries to the blo ck enc o dings U i , and b + O (1) ancil la qubits. Mor e over, { f M i } s − 1 i =0 is a valid quantum instrument. That is, e ach br anch f M i is c ompletely p ositive and P i f M i is tr ac e-pr eserving. Pr o of. Let a anc denote the common ancilla register of the rescaled blo ck enco dings b U i , and write | 0 ⟩ anc := | 0 b ⟩ a anc , b = b + O (1) . (47) W e ﬁrst in tro duce the ideal Stinespring isometry for the instrumen t: V : | ψ ⟩ 7− → s − 1 X i =0 | i ⟩ a 1 | 0 ⟩ anc K i | ψ ⟩ . (48) Here, a 1 denotes an s -dimensional outcome register with computational basis {| i ⟩} s − 1 i =0 , while a anc is a w orkspace ancilla register large enough to accommodate the blo ck-encoding and ampliﬁcation subroutines (see ( 47 )). Measuring a 1 in the computational basis realizes the instrumen t {M i } s − 1 i =0 . Moreo ver, P i K † i K i = I implies V † V = I . Step 1: Common r esc aling of the blo ck enc o dings. T o facilitate the LCU and ampliﬁcation steps b elo w, w e rescale the blo ck enco dings U i of K i to a common normalization constant. F or each i , pad U i with idle ancillas so that all block enco dings act on the same num b er b of ancilla qubits. w e conv ert each U i in to an ( α, b = b + O (1) , ϵ )-blo c k enco ding of K i via a standard common-rescaling gadget: add O (1) ancilla qubits and, for each i , dilute the success amplitude b y p α i /α while routing the remaining amplitude to an orthogonal failure ﬂag. Denoting the resulting unitary for the blo ck enco ding by b U i , it satisﬁes e K i := α ( ⟨ 0 b | ⊗ I ) b U i ( | 0 b ⟩ ⊗ I ) ,    K i − e K i    ≤ ϵ. (49) Step 2: Multiplexing via LCU. W e now construct a blo ck enco ding of the Stinespring-t yp e op erator with e K i deﬁned in ( 49 ): Y : | ψ ⟩ 7− → s − 1 X i =0 | i ⟩ a 1 | 0 ⟩ anc e K i | ψ ⟩ via the individual blo ck enco dings b U i . Deﬁne the pr ep ar e and sele ct oracles b y U prep | 0 ⟩ a 1 := 1 √ s s − 1 X i =0 | i ⟩ a 1 , U sel := s − 1 X i =0 | i ⟩⟨ i | a 1 ⊗ b U i , 36 and set W := U sel ( U prep ⊗ I ) . F or every input state | ψ ⟩ ∈ H S , a direct computation gives W | 0 ⟩ a 1 | 0 ⟩ anc | ψ ⟩ = 1 α √ s s − 1 X i =0 | i ⟩ a 1 | 0 ⟩ anc e K i | ψ ⟩ + | ⊥ ( ψ ) ⟩ , where | ⊥ ( ψ ) ⟩ is orthogonal to span {| i ⟩ a 1 | 0 ⟩ anc ⊗ H S : i = 0 , . . . , s − 1 } . Deﬁning J in | ψ ⟩ := | 0 ⟩ a 1 | 0 ⟩ anc | ψ ⟩ , Π out := s − 1 X i =0 | i ⟩⟨ i | a 1 ⊗ | 0 ⟩⟨ 0 | anc ⊗ I S , w e ha ve Π out W J in = 1 α √ s Y , and hence W is a pro jected unitary enco ding of Y . Step 3: Oblivious amplitude ampliﬁc ation. Recalling P i K † i K i = I and ∥ e K i − K i ∥ ≤ ϵ , w e hav e Y † Y = s − 1 X i =0 e K † i e K i = I + O ( ϵ ) . Hence singular v alues of 1 α √ s Y lie in 1 α √ s [1 − O ( ϵ ) , 1 + O ( ϵ )]. Let U Y := Y ( Y † Y ) − 1 / 2 b e the p olar isometry of Y , whic h satisﬁes U † Y U Y = I and ∥ U Y − Y ∥ = O ( ϵ ). Moreo ver, note that ∥ Y − V ∥ 2 =    s − 1 X i =0 ( e K i − K i ) † ( e K i − K i )    ≤ sϵ 2 , implying ∥ U Y − V ∥ = O ( ϵ ) . (50) No w apply robust oblivious amplitude ampliﬁcation to the pro jected blo c k Π out W J in = 1 α √ s Y . A standard QSVT theorem yields a unitary e V , using O ( α log(1 /ϵ )) queries to W and W † , such that ∥ e V J in − U Y ∥ = O ( ϵ ). Combining this with ( 50 ) giv es ∥ e V J in − V ∥ = O ( ϵ ) , (51) for the ideal Stinespring isometry ( 48 ). Step 4: Instrument implementation cir cuits. W e deﬁne the implemen ted branc h maps by measuring the index register a 1 and tracing out a anc : f M i ( ρ ) := T r anc h  ⟨ i | a 1 ⊗ I  e V ( | 0 , 0 ⟩⟨ 0 , 0 | a 1 , anc ⊗ ρ ) e V †  | i ⟩ a 1 ⊗ I  i . Eac h f M i is completely positive. Moreov er, it holds that s − 1 X i =0 f M i ( ρ ) = T r a 1 , anc h e V ( | 0 , 0 ⟩⟨ 0 , 0 | a 1 , anc ⊗ ρ ) e V † i . 37 Since e V is unitary , the map P i f M i is also trace-preserving. Thus { f M i } s − 1 i =0 is a v alid quantum instrumen t appro ximating the ideal one. Indeed, let ρ ′ := V ρV † , e ρ ′ := e V ( | 0 , 0 ⟩⟨ 0 , 0 | a 1 , anc ⊗ ρ ) e V † . F rom ( 51 ), w e ﬁnd ∥ e ρ ′ − ρ ′ ∥ 1 = O ( ϵ ) . It follows from the con tractivity of the partial trace that s − 1 X i =0 ∥ f M i ( ρ ) − M i ( ρ ) ∥ 1 ≤ ∥ e ρ ′ − ρ ′ ∥ 1 = O ( ϵ ) . This prov es the theorem. C Implemen tation of the sup erop erator R T o implement the rejection branc h R [ · ] = K ( · ) K † with the Kraus op erator K = q √ σ β ( I − O ) √ σ β σ − 1 / 2 β , O := T ′† [ I ] = K † 1 K 1 + K † 2 K 2 , (52) w e emplo y the implemen tation techniques dev elop ed in [ GCDK26 , App endix C] for discrete-time quan tum Gibbs samplers. The key s tructural condition required is that O b e quasi-lo c al in ener gy with resp ect to H . T o mak e this precise, recall that any op erator X ∈ B ( H ) admits a Bohr-frequency decomposition X = X ν X ν , e iH t X ν e − iH t = e iν t X ν , where ν = E k − E j ranges ov er the Bohr frequencies of H = P k E k | k ⟩⟨ k | . Deﬁnition 26 (Quasi-lo cal in energy) . An op erator X ∈ B ( H ) is (Ω , ϵ ′ )-quasi-lo cal in energy with resp ect to H if there exists an op erator e X such that ∥ X − e X ∥ ≤ ϵ ′ and e X ν = 0 for all | ν | ≥ Ω, or equiv alen tly , ⟨ E j | e X | E k ⟩ = 0 whenev er | E j − E k | ≥ Ω . W e use the following results from [ GCDK26 , Prop osition 5 and Lemma 12]. Prop osition 27. L et ϵ ∈ (0 , 1 / 4] , s = Θ(log( β ∥ H ∥ )) , and α ≥ ∥ H ∥ . Supp ose ∥ O ∥ ≤ 10 − 3 s 2 and O is (Ω , ϵ ′ ) -quasi-lo c al in ener gy with r esp e ct to H , wher e Ω = 1 10 ⌊ log 2 (1 /ϵ ) ⌋ , ϵ ′ = O  ϵ s 2 p olylog(1 /ϵ )  . (53) Then one c an implement an ϵ -appr oximate blo ck-enc o ding of 1 2 q √ σ β ( I − O ) √ σ β σ − 1 / 2 β using O ( αβ p olylog ( αβ /ϵ )) queries to a blo ck-enc o ding of H/α , O (p olylog(1 /ϵ )) queries to a blo ck- enc o ding of O , and O (p olylog ( αβ /ϵ )) ancil lary qubits. 38 In our setting, the small-norm condition on O in ( 52 ) is satisﬁed by the choice c = O (1 / log( β ∥ H ∥ )) , in our constructions ( 20 ) and ( 22 ), since O = T ′† [ I ] ≤ 3 c 2 I by ( 23 ). It therefore remains to v erify the quasi-lo cality in energy condition. Lemma 28 (T runcated Gaussian ﬁltered op erators) . L et χ ∈ C ∞ c ( R ) b e a smo oth cutoﬀ function satisfying 0 ≤ χ ≤ 1 , χ ( x ) = 1 on | x | ≤ 1 2 , supp χ ⊂ [ − 1 , 1] . F or Ω 0 > 0 , deﬁne χ Ω 0 ( ν ) := χ ( ν / Ω 0 ) . F or g ∈ { f , ˜ f } , deﬁne the trunc ate d op er ator and its err or by A (Ω 0 ) g := X ν χ Ω 0 ( ν ) b g ( ν ) A ν , (54) Then A (Ω 0 ) g has Bohr-fr e quency supp ort c ontaine d in [ − Ω 0 , Ω 0 ] . Mor e over, let ∥ A ∥ ≤ 1 and τ = Ω( β ) . It holds that with c f = 1 / 2 and c ˜ f = 1 / 4 , A g − A (Ω 0 ) g = O ((1 + τ Ω 0 ) 3 exp  − c g τ 2 Ω 2 0  ) g ∈ { f , ˜ f } . Pr o of. Deﬁne m g , Ω 0 ( ν ) :=  1 − χ Ω 0 ( ν )  b g ( ν ) , η g , Ω 0 ( t ) := 1 2 π Z R m g , Ω 0 ( ν ) e iν t d ν. Since A g = R R g ( t ) e iH t Ae − iH t d t = P ν b g ( ν ) A ν , we ha ve E (Ω 0 ) g := A g − A (Ω 0 ) g = Z R η g , Ω 0 ( t ) e iH t Ae − iH t d t, and therefore ∥ E (Ω 0 ) g ∥ ≤ ∥ A ∥ ∥ η g , Ω 0 ∥ L 1 ≤ ∥ η g , Ω 0 ∥ L 1 . W e shall use the following basic estimate: ∥ η ∥ L 1 ≤ 1 π  ∥ m ∥ L 1 + ∥ m ′′ ∥ L 1  , (55) where m ∈ W 2 , 1 ( R ) and η ( t ) = 1 2 π R m ( ν ) e iν t d ν . Indeed, for | t | ≤ 1 we hav e | η ( t ) | ≤ 1 2 π ∥ m ∥ L 1 , while for | t | > 1, integrating by parts twice giv es | η ( t ) | ≤ 1 2 π t 2 ∥ m ′′ ∥ L 1 . Integrating ov er | t | ≤ 1 and | t | > 1 yields the b ound. W e claim that for j = 0 , 1 , 2 and τ = Ω( β ), it holds that for some constant c g , b g ( j ) ( ν ) = O ( τ j (1 + τ | ν | ) j e − c g τ 2 ν 2 ) , g ∈ { f , ˜ f } . Center e d Gaussian. F or g = f , w e ha v e b f ( ν ) = e − τ 2 ν 2 / 2 , and for j = 0 , 1 , 2 there holds | b f ( j ) ( ν ) | ≤ τ j (1 + τ | ν | ) j e − τ 2 ν 2 / 2 . 39 Shifte d Gaussian. F or g = ˜ f , recall b ˜ f ( ν ) = e β 2 / (8 τ 2 ) exp  − τ 2 2 ( ν − µ ) 2  , µ := β 2 τ 2 . Using ( ν − µ ) 2 ≥ ν 2 2 − µ 2 , we obtain | b ˜ f ( ν ) | ≤ e β 2 / (8 τ 2 ) e τ 2 µ 2 / 2 e − τ 2 ν 2 / 4 = e β 2 / (4 τ 2 ) e − τ 2 ν 2 / 4 . Since τ = Ω( β ), the prefactor e β 2 / (4 τ 2 ) is O (1). The deriv atives satisfy b ˜ f ′ ( ν ) = − τ 2 ( ν − µ ) b ˜ f ( ν ) , b ˜ f ′′ ( ν ) =  τ 4 ( ν − µ ) 2 − τ 2  b ˜ f ( ν ) , hence for j = 0 , 1 , 2 there holds | b ˜ f ( j ) ( ν ) | = O ( τ j (1 + τ | ν | ) j e − τ 2 ν 2 / 4 ) . Since m g , Ω 0 ( ν ) = (1 − χ Ω 0 ( ν )) b g ( ν ), a direct computation giv es m ′′ g , Ω 0 = (1 − χ Ω 0 ) b g ′′ − 2 χ ′ Ω 0 b g ′ − χ ′′ Ω 0 b g . The deriv ativ es χ ′ Ω 0 and χ ′′ Ω 0 are supp orted in { Ω 0 / 2 ≤ | ν | ≤ Ω 0 } and satisfy ∥ χ ′ Ω 0 ∥ ∞ = O (Ω − 1 0 ) , ∥ χ ′′ Ω 0 ∥ ∞ = O (Ω − 2 0 ) . Com bining the deriv ativ e bounds with the supp ort prop erties yields, for c, C > 0, ∥ m g , Ω 0 ∥ L 1 + ∥ m ′′ g , Ω 0 ∥ L 1 ≤ C (1 + τ Ω 0 ) 3 e − cτ 2 Ω 2 0 . Applying the F ourier L 1 estimate ( 55 ) completes the proof. Lemma 29 (Quasi-lo cality in energy of T ′† [ I ]) . L et K 1 and K 2 b e deﬁne d as in ( 20 ) and ( 22 ) , r esp e ctively, and O = T ′† [ I ] b e given in ( 52 ) . Given ϵ ∈ (0 , 1 / 4] , let Ω and ϵ ′ b e as in ( 53 ) r e quir e d by Pr op osition 27 . Assume that the width p ar ameter τ of f satisﬁes τ = max ( Ω( β ) , Ω k Ω r log  k c 2 ϵ ′  !) with k = Θ(log (1 /ϵ ′ )) , and u satisﬁes | u | ≤ 1 / (2 ∥ A ˜ f ∥ ) fr om The or em 24 . Then, O is (Ω , ϵ ′ ) -quasi- lo c al in ener gy with r esp e ct to H . Equivalently, ther e exists an op er ator e O such that e O ν = 0 for al l | ν | ≥ Ω , ∥ O − e O ∥ ≤ ϵ ′ . Pr o of. W e write O = c 2 ( I + uA f ) + c 2 X † X , X := q I + uA ˜ f . Since ∥ A ˜ f ∥ = O (1) for τ = Ω( β ), choosing | u | ≤ 1 / (2 ∥ A ˜ f ∥ ) ensures r = | u | ∥ A ˜ f ∥ ≤ 1 / 2. Let P k ( z ) := P k j =0  1 / 2 j  z j b e the degree- k T a ylor polynomial of √ 1 + z with ∥ X − P k ( uA ˜ f ) ∥ ≤ r k +1 1 − r . 40 With k = Θ(log(1 /ϵ ′ )), we can ensure ∥ X † X − P k ( uA ˜ f ) † P k ( uA ˜ f ) ∥ = O ( ϵ ′ /c 2 ) , (56) since b oth ∥ X ∥ and ∥ P k ( uA ˜ f ) ∥ are O (1). Therefore, if w e deﬁne O ( k ) := c 2 ( I + uA f ) + c 2 P k ( uA ˜ f ) † P k ( uA ˜ f ) , then ∥ O − O ( k ) ∥ = O ( ϵ ′ ) . (57) Next, let Ω 0 := Ω 2 k , and deﬁne the truncated op erators A (Ω 0 ) f and A (Ω 0 ) ˜ f as in ( 54 ). By Lemma 28 , we choose τ = Ω  k Ω q log  k c 2 ϵ ′   large enough suc h that max g ∈{ f , ˜ f } ∥ A g − A (Ω 0 ) g ∥ ≤ min n 1 4 | u | , Θ  ϵ ′ c 2  o . (58) Set B := uA ˜ f and e B := uA (Ω 0 ) ˜ f . Then ∥ B ∥ ≤ 1 2 and ∥ e B ∥ ≤ 3 4 . The telescoping identit y giv es, for eac h j ≥ 1, ∥ B j − e B j ∥ ≤ j  3 4  j − 1 ∥ B − e B ∥ = j  3 4  j − 1 | u | ∥ A ˜ f − A (Ω 0 ) ˜ f ∥ . whic h implies, by P j ≥ 1 |  1 / 2 j  | j ( 3 4 ) j − 1 < ∞ and ( 58 ), ∥ P k ( B ) − P k ( e B ) ∥ ≤ k X j =1      1 / 2 j      ∥ B j − e B j ∥ = O  ϵ ′ c 2  . Noting ∥ P k ( B ) ∥ , ∥ P k ( e B ) ∥ = O (1), w e ha v e ∥ P k ( B ) † P k ( B ) − P k ( e B ) † P k ( e B ) ∥ = O  ϵ ′ c 2  . (59) W e now deﬁne e O := c 2  I + uA (Ω 0 ) f  + c 2 P k  uA (Ω 0 ) ˜ f  † P k  uA (Ω 0 ) ˜ f  . and sho w that it is the desired appro ximation of O . Indeed, since A (Ω 0 ) g has Bohr-frequency supp ort in [ − Ω 0 , Ω 0 ], the ﬁrst term has supp ort in [ − Ω 0 , Ω 0 ] and P k ( uA (Ω 0 ) ˜ f ) † P k ( uA (Ω 0 ) ˜ f ) has supp ort in [ − 2 k Ω 0 , 2 k Ω 0 ] = [ − Ω , Ω]. Hence e O has Bohr-frequency supp ort in [ − Ω , Ω]. F or the appro ximation error, by ( 58 ) and ( 59 ), w e hav e ∥ e O − O ( k ) ∥ ≤ c 2 | u | ∥ A f − A (Ω 0 ) f ∥ + c 2 ∥ P k ( B ) † P k ( B ) − P k ( e B ) † P k ( e B ) ∥ = O ( ϵ ′ ) . Com bining it with ( 57 ), w e hav e ∥ O − e O ∥ = O ( ϵ ′ ), and hence O is (Ω , ϵ ′ )-quasi-lo cal in energy with resp ect to H . 41 References [A CL Y24] Dong An, Andrew M. Childs, Lin Lin, and Lexing Ying, L aplac e tr ansform b ase d quan- tum eigenvalue tr ansformation via line ar c ombination of hamiltonian simulation , 2024. [AS16] Noga Alon and Jo el H. 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Single-Trajectory Gibbs Sampling for Non-Commuting Observables

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