quant-ph 2026-02-25 1

Learning Quantum Data Distribution via Chaotic Quantum Diffusion Model

Generative models for quantum data pose significant challenges but hold immense potential in fields such as chemoinformatics and quantum physics. Quantum denoising diffusion probabilistic models (QuDDPMs) enable efficient learning of quantum data dis…

Authors: Quoc Hoan Tran, Koki Chinzei, Yasuhiro Endo

Learning Quantum Data Distribution via Chaotic Quantum Diffusion Model
Learning Quan tum Data Distribution via Chaotic Quan tum Diffusion Mo del Quo c Hoan T ran, ∗ Koki Chinzei, Y asuhiro Endo, and Hirotak a Oshima Quantum L ab or atory, F ujitsu R ese ar ch, F ujitsu Limite d, Kawasaki, Kanagawa 211-8588, Jap an (Dated: F ebruary 26, 2026) Generativ e models for quantum data pose significant challenges but hold immense p oten tial in fields suc h as c hemoinformatics and quan tum physics. Quan tum denoising diffusion probabilistic mo dels (QuDDPMs) enable efficien t learning of quan tum data distributions b y progressiv ely scram- bling and denoising quantum states; ho wev er, existing implementations typically rely on circuit- based random unitary dynamics that can be costly to realize and sensitiv e to con trol imp erfections, particularly on analog quantum hardware. W e prop ose the chaotic quantum diffusion mo del, a framew ork that generates pro jected ensembles via chaotic Hamiltonian time ev olution, providing a flexible and hardware-compatible diffusion mec hanism. Requiring only global, time-indep endent con trol, our approach substan tially reduces implemen tation o verhead across div erse analog quan tum platforms while ac hieving accuracy comparable to QuDDPMs. This method improv es trainability and robustness, broadening the applicability of quantum generative mo deling. INTR ODUCTION Generativ e mo dels aim to synthesize diverse data by learning their underlying probability distributions and pla y a central role in data understanding, sim ulation, and discov ery . Quan tum circuits are capable of gen- erating classically intractable probability distributions, motiv ating the use of quantum systems as generative mo dels that may outp erform their classical counterparts in certain regimes. Along this line, a v ariety of quan- tum generativ e mo dels hav e been prop osed for classical data generation, including quantum generative adv ersar- ial netw orks (QuGANs) [ 1 – 3 ], quantum v ariational au- to encoders (QV AEs) [ 4 , 5 ], tensor-netw ork-based mo d- els [ 6 ], and diffusion-based quantum models [ 7 , 8 ]. De- spite these developmen ts, a clear empirical adv an tage o ver classical generativ e mo dels has yet to be established. More recen tly , growing attention has shifted tow ard quan tum machine learning (QML) for quantum data, where the data themselves originate from quantum sys- tems [ 9 ]. Unlike classical data suc h as text or images, quan tum data inherently enco de quantum correlations, en tanglement, and measuremen t statistics, posing funda- men tal challenges that cannot be addressed by classical generativ e mo dels alone. Efficien t generation of quan- tum data is essen tial for adv ancing our understanding of quantum many-bo dy systems and for applications in c hemistry , biology , and materials science. Since quantum data generation intrinsically relies on quan tum resources, quan tum-native generative mo dels are a natural and nec- essary choice. Existing quantum generative mo dels such as QuGANs and QV AEs can b e used to prepare individual target quan tum states [ 10 – 12 ], but they are generally ineffi- cien t at learning and generating ensembles of quantum states [ 13 ]. This limitation arises primarily from the need to train deep v ariational quantum circuits (V QCs), whic h suffer from optimization challenges such as barren plateaus [ 14 ]. The quantum denoising diffusion proba- bilistic mo del (QuDDPM) [ 15 ] addresses these issues by extending the classical diffusion paradigm to quantum data. QuDDPM learns a sequence of intermediate distri- butions that interpolate smo othly b et w een a structured target distribution E 0 and a maximally scrambled distri- bution, using quantum scrambling for the forw ard diffu- sion pro cess and measurement-enabled VQCs for bac k- w ard denoising. This lay erwise training strategy im- pro ves trainabilit y and mitigates barren plateaus. How- ev er, existing realizations of QuDDPM rely on circuit- based scrambling using random unitary circuits (R UCs), whic h require fine-grained spatio-temp oral con trol and substan tial circuit depth. These requiremen ts p ose signif- ican t challenges for implementation, particularly on ana- log quan tum platforms where con trol is t ypically global and time-indep endent. In this work, we prop ose a chaotic quantum diffusion mo del that replaces circuit-based scram bling with chaotic Hamiltonian time ev olution. By generating pro jected en- sem bles through quan tum c haos, our approach pro vides a flexible and hardware-compatible diffusion mechanism that eliminates the need for explicit R UCs, enabling ef- ficien t implementation across a broad range of quantum hardw are. QUANTUM DENOISING DIFFUSION PR OBABILISTIC MODEL (QUDDPM) W e address the problem of learning quan tum data dis- tributions and explain the idea of the QuDDPM mo del in the literature [Fig. 1 (a)]. Consider a training dataset S of N indep endent quan tum states dra wn from an un- kno wn probability distribution E 0 . A generative model is c haracterized b y a parameterized probabilit y distribution E θ , from which samples can be drawn. If E θ is realized through V QCs, θ corresp onds to the adjustable param- eters of these circuits. The training goal is to obtain a distribution E θ that closely approximates E 0 . W e opti- 2 FIG. 1. (a) The general sc heme of quantum denoising diffusion probabilistic mo del. (b) The implementation of the random unitary circuit diffusion (RUCD) mo del. (c) The chaotic quantum diffusion mo del in our prop osal. mize the distance D ( E θ , E 0 ) suc h as the Maximum Mean Discrepancy (MMD) and W asserstein distance b etw een these distributions. Since D ( E θ , E 0 ) cannot b e computed directly , w e tak e a dataset S θ sampling from E θ and com- pute the distance D ( S , S θ ). In the inference phase, w e fix the trained θ and generate new states | ψ ⟩ ∼ E θ . Unlik e prior quan tum models that focus on generat- ing av erage states such as fully mixed states, QuDDPM targets entire distributions of states, enabling the gener- ation of correlated or structured quan tum data. QuD- DPM builds on the success of classical DDPMs in tasks suc h as image generation but extends them to quantum data. Classical DDPMs use Gaussian noise addition in a forward process to gradually corrupt data tow ard a noise distribution (e.g., an isotropic Gaussian), follo wed b y learned denoising in the reverse pro cess. QuDDPM replaces Gaussian noise with scrambling RUCs in the forw ard pro cess, applying random unitaries that evolv e quan tum states tow ard a near–Haar-random distribu- tion. This choice is more natural for quantum systems, as Gaussian noise do es not directly corresp ond to physically v alid unitary dynamics. In the backw ard pro cess, QuDDPM employs measuremen t-enabled denoising using VQCs with ancilla qubits and pro jective measurements. The ancilla qubits are en tangled with the system qubits via con trolled, parameterized op erations, and subsequent measure- men ts collapse the ancilla states, effectively denoising b y pro jecting the system onto subspaces that approximate the target distribution. These measurement-induced, non-unitary operations are essen tial for breaking the constrain ts of purely unitary quantum circuits and enabling flexible generative mo deling. This con trasts with fully unitary quantum mo dels, which are limited in expressivity when learning inheren tly non-unitary pro cesses such as noise. F orward diffusion pro cess In the forward diffusion pro cess using RUCs, K random unitary gates U ( j ) 1 , . . . , U ( j ) K are applied to eac h sample | ψ (0) j ⟩ ( j = 1 , . . . , N ) in the train- ing dataset S [Fig. 1 (b)]. This evolv es the ensem- ble S k = n | ψ ( k ) j ⟩ = Q k l =1 U ( j ) l | ψ (0) j ⟩ o j to ward a Haar- random states ensemble ov er the Hilb ert space. W e de- fine this scheme as RUCs diffusion (RUCD). Assuming the execution time for eac h U ( j ) k is τ u , the execution time to generate eac h S k is τ u N k . Therefore, the RUCD re- quires N K random unitary gates with τ u N P K k =1 k = τ u N K ( K + 1) / 2 execution time to generate all S k . The forward diffusion pro cess do es not require exact Haar randomness. Recen t studies ha v e sho wn that shal- lo w random circuits can efficiently approximate unitary k -designs with depth scaling p olynomially or even loga- rithmically in system size for small k [ 16 ]. Such approx- imate designs are sufficien t to induce rapid scrambling of lo cal observ ables and correlations, making them suit- able for diffusion-based generativ e mo deling. QuDDPM’s implemen tation relies on approximate scrambling rather than high-fidelity Haar sampling. Nevertheless, ev en shallo w circuit constructions for approximate designs re- quire explicit gate-lev el con trol and circuit compilation, whic h can b e costly or infeasible on analog quan tum plat- forms. This motiv ates alternativ e diffusion mechanisms that naturally generate effective scrambling without the need for circuit-based random unitaries. Bac kward denoising process The bac kward pro cess starts with an ensem ble ˜ S K = {| ˜ ψ ( K ) j ⟩} j sampled from Haar-random states and reduces noise step by step. In practice, since it is difficult to sample from a complete Haar-random state distri- bution, ˜ S K = {| ˜ ψ ( K ) j ⟩} j are sampled from the pro d- 3 uct of single-qubit Haar-random states. As depicted in Fig. 1 (a), the denoising step applies a parameterized uni- tary V k = V ( θ k ) to the data system D (input | ˜ ψ ( k ) j ⟩ ) and n a ancilla qubits in the ancilla system A ( | 0 ⟩ A ), follow ed b y pro jectiv e measuremen ts in the computational basis on A , yielding the state | ˜ ψ ( k − 1) j ⟩ in D . Assuming the measuremen t outcome z ( k ) j is obtained on the ancilla, this op eration is formulated as Φ ( k ) j ( | ˜ ψ ( k ) j ⟩ ) = ( I D ⊗ Π A ) V k | ˜ Ψ ( k ) j ⟩ q ⟨ ˜ Ψ ( k ) j | V † k ( I D ⊗ Π A ) V k | ˜ Ψ ( k ) j ⟩ (1) = | ˜ ψ ( k − 1) j ⟩ ⊗ | z ( k ) j ⟩ A , (2) where Π A = | z ( k ) j ⟩ ⟨ z ( k ) j | A and | ˜ Ψ ( k ) j ⟩ = | ˜ ψ ( k ) j ⟩ ⊗ | 0 ⟩ A . T raining inv olves K cycles with a la y erwise sc heme. At the cycle ( K − k + 1) (with k = K, . . . , 1), the forward diffusion with U ( j ) 1 to U ( j ) k − 1 generates the noisy ensemble S k − 1 = n | ψ ( k − 1) j ⟩ o j . Parameters of V ( θ k ) are optimized to make the denoised ensemble ˜ S k − 1 = n | ˜ ψ ( k − 1) j ⟩ o j ap- pro ximate S k − 1 via the minimization of the cost function D ( S k − 1 , ˜ S k − 1 ). After optimization, the parameters θ k are fixed for use in the next cycle to optimize θ k − 1 . This la yerwise training approach divides the original training problem into K manageable sub-tasks, ensuring con ver- gence for incremental distribution transitions and miti- gating issues such as barren plateaus [ 14 ]. The cost function in QuDDPM measures the similarity b et w een tw o quan tum state ensembles using a symmetric, p ositiv e definite quadratic kernel κ ( | µ ⟩ , | ϕ ⟩ ). This kernel can b e defined by the state fidelity computed via the SW AP test (Fig. 2 ) or directly derived from the classical shado ws kernel [ 17 , 18 ]. W e consider tw o cost functions D MMD and D W ass , corresponding to MMD distance and 1-W asserstein distance based on the state fidelity . The MMD distance b et ween tw o state ensembles X = {| µ i ⟩} and Y = {| ψ j ⟩} is defined as D MMD ( X , Y ) = ¯ κ ( X , X ) + ¯ κ ( Y , Y ) − 2¯ κ ( X , Y ) , (3) where ¯ κ ( X , Y ) = E | µ ⟩∈X , | ϕ ⟩∈Y [ κ ( | µ ⟩ , | ϕ ⟩ )]. The 1-W asserstein distance is further presented as an enhancemen t in the situation where the MMD distance is not feasible to distinguish tw o state ensembles. Given normalized κ (i.e., κ ( | ϕ ⟩ , | ϕ ⟩ ) = 1 ∀ | ϕ ⟩ ), the pairwise cost matrix C = ( C i,j ) ∈ R |X |×|Y | is computed as C i,j = 1 − κ ( | µ i ⟩ , | ψ j ⟩ ). The 1-W asserstein distance is calculated via the formulation of the optimal transp ort problem into a linear programming pro cedure to find the optimal transp ort plan P = ( P i,j ) ∈ R |X |×|Y | : D W ass ( X , Y ) = min P X i,j P i,j C i,j , (4) s.t. P 1 |Y | = a , P ⊤ 1 |X | = b , P ≥ 0 . (5) Here, 1 |X | and 1 |Y | are all-ones vectors with size |X | and |Y | , resp ectively , and a ∈ R |X | and b ∈ R |Y | are the probabilit y vectors histogram corresp onding to X and Y . Normally , w e set uniform histograms as a = 1 |X | 1 |X | and b = 1 |Y | 1 |Y | . PR OPOSAL: CHAOTIC QUANTUM DIFFUSION R UCD demands significant computational ov erhead in designing sequences of random gates, rendering it unsuit- able for man y quantum systems, particularly analog plat- forms such as Rydb erg atom arrays and ultracold atoms in optical lattices. These platforms excel in simulating con tinuous-time dynamics via global Hamiltonians but lac k the fine-grained, time-dep endent control needed for arbitrary gate sequences. Implemen ting R UCD on ana- log hardware w ould require digitizing the pro cess through T rotterization or pulse shaping, whic h in tro duces appro x- imation errors and increases susceptibility to noise from en vironmental couplings. W e address this challenge by adopting the pro jected ensem ble framework [ 19 – 21 ], whic h utilizes a single c haotic many-bo dy wa ve function to generate a random ensem ble of pure states on a subsystem. In this frame- w ork, pro jective measurements are p erformed on the larger subsystem of a bipartite state undergoing quan- tum chaotic ev olution. This process yields a set of pure states on the smaller subsystem, accompanied by their resp ectiv e Born probabilities. Collectively , these states form the pro jected ensem ble, whic h con verges to a state design when the measured subsystem is sufficien tly large. A c haotic Hamiltonian in this con text is c haracterized b y non-in tegrability , ergo dic dynamics, sp ectral prop erties resem bling those of random matrix theory , and strong en- tanglemen t generation. Recent works hav e further clar- ified the role of symmetry in the Hamiltonian and mea- suremen t structure in the emergence of state designs from pro jected ensembles [ 22 ]. Our approac h offers significan t adv an tages o ver R UCD, as it requires only global and time-indep enden t control. Consequen tly , it is more accessible and adaptable to a wide range of quan tum systems, including analog plat- forms, where precise gate sequences are c hallenging to implemen t. Pro jected Ensem ble W e consider a man y-b ody system partitioned into a subsystem M (with n m qubits) and its complemen t F (with n f qubits). Given a generator state | Φ ⟩ on the total system M + F , which is produced by chaotic ev olution, we p erform lo cal measurements on F , typ- ically in the computational basis. This yields differ- 4 en t pure states | Φ M ( z F ) ⟩ on M , eac h corresp onding to a distinct measurement outcome z F , which are bit- strings of the form, for example, z F = 001 . . . 010. The collection of these states, together with probabilities p ( z F ) = ∥ ( I M ⊗ ⟨ z F | ) | Φ ⟩ ∥ 2 , forms the pro jected ensem- ble on M : {| Φ M ( z F ) ⟩ , p ( z F ) } z F . The pro jected ensem- ble provides a full description of the total system state as | Φ ⟩ = P z F p p ( z F ) | Φ M ( z F ) ⟩ ⊗ | z F ⟩ . W e note that the ensem ble of pure states is not just a density matrix. The densit y matrix represents the a verage mixed state o ver the distribution, capturing first-order statistics such as exp ectation v alues of observ ables. Therefore, different ensem bles can lead to the same density matrix. These ensem bles can b e distinguished through their higher mo- men ts of the observ able exp ectation v alues. The chaotic dynamics scramble quan tum information, resulting in univ ersal correlations and randomness within the subsystem. F or a case of infinite-temp erature ther- malization, with sufficiently large n f = Ω( k n m ) and the generator state | Φ ⟩ obtained by quenched time ev o- lution of chaotic Hamiltonians, the pro jected ensem ble appro ximates k -design of Haar-random states [ 19 , 20 ]. This con vergence is quan tified b y the trace distance ∆ ( k ) = 1 2 ∥ ρ ( k ) E − ρ ( k ) H aar ∥ 1 → 0 for the k -th momen t op era- tors ρ ( k ) E and ρ ( k ) H aar calculated from the pro jected ensem- ble and the Haar-random states ensemble, resp ectively . Mathematically , this relies on the eigenstate thermaliza- tion h yp othesis and the concentration of measure: the c haotic evolution scram bles quantum information uni- v ersally , making higher momen ts con v erge to the Haar measure. Inspired by this framew ork, w e prop ose tw o schemes in whic h the diffusion is implemented through chaotic Hamiltonian evolution [Fig. 1 (c)]. Cum ulative Time Ev olution Diffusion (CTED) F or each | ψ (0) j ⟩ ∈ S 0 on M , we implemen t a diffusion pro cess to obtain the k -step state | ψ ( k ) j ⟩ ∈ S k . The initial state | x ( k ) j ⟩ in F is randomly sampled from the ensemble {| x ⟩ , q ( x ) } x ∈{ 0 , 1 } n f of com putational basis states with probabilit y q ( x ). Inspired from Ref. [ 21 ], we introduce q ( x ) (uniform distribution in our exp eriments) here to increase classical randomness injected during the proto- col, which is conv erted in to quantum randomness of the resulting state ensemble. The input | ψ (0) j ⟩ ⊗ | x ( k ) j ⟩ of the system ( M + F ) is then ev olved cumulativ ely under the same unitary e − iH k ∆ t with ∀ j , where H is a fixed c haotic Hamiltonian [ 20 ]. Subsequen tly , a random lo- cal measurement is p erformed on F , yielding the mea- suremen t record z ( k ) j with the probability p x ( z ( k ) j ) = q ( x ( k ) j )     I M ⊗ ⟨ z ( k ) j |  e − iH k ∆ t  | ψ (0) j ⟩ ⊗ | x ( k ) j ⟩     2 , and X X FIG. 2. The sc hematic circuit to compute the fidelit y betw een the forward state | ψ ( k − 1) j ⟩ and the denoised state | ˜ ψ ( k − 1) j ⟩ using the SW AP test. the resulting state | ψ ( k ) j ⟩ on M . In this case, we ob- tain the classic al ly-enhanc e d pro jected ensem ble E ( k ) = { q ( x ( k ) j ) p x ( z ( k ) j ); | ψ x ( z ( k ) j ) ⟩} j . Since the execution time for e − iH k ∆ t is generally con- sidered scaling with k , we can assume the execution time for e − iH k ∆ t is k τ c . This CTED requires K unitaries and τ c N P K k =1 k = τ c N K ( K + 1) / 2 execution time to gener- ate all S k . In Fig. 2 , we depict the schematic circuit to compute the state fidelit y b etw een the forward state | ψ ( k − 1) j ⟩ and the denoised state | ˜ ψ ( k − 1) j ⟩ can be computed using the SW AP test. The SW AP test consists of tw o Hadamard gates and a con trolled-swap gate applied on 2 n m + 1 qubits. The probability of measure 0 on the SW AP reg- ister is p (0) = 1 2 + 1 2    ⟨ ψ ( k − 1) j | ˜ ψ ( k − 1) j ⟩    2 , which directly deriv es the fidelity . A post-selection proto col is applied to the pro jective measuremen ts in the forward and back- w ard pro cesses, ensuring | ψ ( k − 1) j ⟩ and | ˜ ψ ( k − 1) j ⟩ remain consisten t across SW AP test measurements. Rep eated Time Evolution Diffusion (R TED) Unlik e the CTED, we consider the input of ( M + F ) as | ψ ( k − 1) j ⟩ ⊗ | x ( k ) j ⟩ , where | ψ ( k − 1) j ⟩ ∈ S k − 1 , then evolv e this state under the unitary e − iH ∆ t with ∀ j . This pro cess is then rep eated for k = 1 , . . . , K . Assuming the execution time for e − iH ∆ t is τ r , to generate each S k , w e need to rep eat e − iH ∆ t for k times, making the execution time is k τ r . Therefore, this R TED requires only one unitary but τ r N P K k =1 k = τ r N K ( K + 1) / 2 execution time to generate all S k . Univ ersal Ensemble In the CTED and R TED schemes, the generator state at eac h diffusion step is generally not at infinite tem- p erature. As a result, the forward diffusion op erates in a 5 finite-temp erature regime where conv ergence to the Haar measure is not exp ected [ 20 ]. In CTED, longer evolution times ( k ∆ t ) allow more scrambling. How ev er, b ecause the dynamics start from finite-energy states, the resulting pro jected ensembles remain constrained by energy con- serv ation and saturate at a finite-temp erature univ ersal distribution. In R TED, resets on diffused states preserve some initial structure across steps, further deviating from full Haar scram bling. If the target distribution E 0 w ere al- ready Haar-random, the pro jected ensembles would con- v erge to approximate k -designs, but in the general c ase, the target distributions are often lo w-en tropy , making the finite-temp erature-lik e distribution. Recen t w ork has shown that, under chaotic many-bo dy dynamics at finite energy density , pro jected ensembles generically con v erge to Scro oge ensem bles [ 23 , 24 ]. These ensem bles are universal in the sense that they repro duce ETH-predicted mome n ts for lo cal observ ables across dif- feren t c haotic Hamiltonians, yet remain maximally con- strained by conserved quantities such as energy [ 25 ]. F rom an information-theoretic p ersp ective, Scro oge en- sem bles can b e understo od as maximum-en trop y distri- butions sub ject to fixed macroscopic constraints, em- b odying the minimal amount of randomness required by the dynamics rather than full Haar randomness. The Scro oge-lik e nature of the diffuse ensem ble af- fects bac kward denoising but do es not hinder trainabil- it y . In con trast to RUCD, where diffusion tow ard fully Haar-random states can lead to barren-plateau behav- ior in deep v ariational quantum circuits, the energy- dep enden t structure retained in finite-temp erature diffu- sion may provide a smoother learning landscape. Our n umerical results demonstrate that, despite deviating from Haar scram bling, CTED and R TED achiev e gen- erativ e accuracy comparable to RUCD while op erating in a more physically realistic and information-efficien t diffusion regime. RESUL TS W e conduct n umerical exp erimen ts using three illustra- tiv e datasets: synthetic distributions of clustered quan- tum states, circular quantum states, and a quantum dis- tribution derived from a chemistry dataset. W e simulate the quantum circuits with the T ensorCircuit library [ 26 ], and rely on JAX [ 27 ] for automatic differen tiation to sup- p ort gradient-based optimization. The circuit parame- ters are initialized uniformly within [ − π , π ], and opti- mization is p erformed using the Adam algorithm with a learning rate of 0 . 001. Source co de to repro duce these exp erimen ts is av ailable in the GitHub rep ository [ 28 ]. F or CTED and R TED, the dynamics are go v erned b y a one-dimensional mixed-field Ising Hamiltonian with op en b oundary conditions, defined on the total system com- prising n m + n f sites as H = P n m + n f j =1  h x σ x j + h y σ y j  + J P n m + n f − 1 j =1 σ x j σ x j +1 . Here, σ µ j (with µ = x, y , z ) denotes the Pauli operators at site j , J represents the interaction strength, and h x and h y are the strengths of the longi- tudinal and transverse magnetic fields, resp ectiv ely . In the presence of a non-zero longitudinal field ( h x  = 0), the Hamiltonian exhibits ergo dic behavior [ 20 ], with its eigen v alues and eigenv ectors conforming to the predic- tions of the ETH [ 29 , 30 ]. T o mo del the non-integrable regime, we adopt h x = 0 . 8090, h y = 0 . 9045, and J = 1 . 0. The time evolution is discretized with a time step of ∆ t = 0 . 02 ov er K diffusion steps. F or RUCD, we adopt the fast scram bling circuits from Ref. [ 15 ], applying Q k l =1 U ( j ) k to each initial state | ψ (0) j ⟩ . Here, U ( j ) k = Ω k ( s ( j ) k ) W k ( g ( j ) k ), where W k ( g ( j ) k ) implemen ts single-qubit rotations as W k ( g ( j ) k ) = N n m q =1 e − ig ( j ) k, 3 q − 1 Z q 2 e − ig ( j ) k, 3 q − 2 Y q 2 e − ig ( j ) k, 3 q − 3 Z q 2 , and an entan- gling la yer Ω k ( s ( j ) k ) applies ZZ rotations across all qubit pairs as Ω k ( s ( j ) k ) = Q q 1

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