Distinguishing Short Quantum Computations

DISTINGUISHING SHOR T QUANTUM COMPUT A TIONS BILL ROSGEN 1 1 Institute for Quantum Computing and S chool of Compu t er Science, U niversi ty of W aterl oo E-mail addr ess : wrosgen@iq c.ca Abstra ct. Distinguishing logarithmic depth quan tum circuits on mixed states is shown to b e complete for QIP, the class of problems having quantum interactiv e proof systems. Circuits in this mo del can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of qu antum algori thms. The distin- guishabilit y problem is also complete for QIP on constant dep th circuits containing the unbound ed fan-out gate. These results are show n by reducing a QIP-complete problem to a logarithmic d epth ve rsion of itself using a parallelizatio n technique. 1. In tro duction Muc h of the d ifficult y in implemen ting quan tum algorithms in pr actice is that qu bits quic kly decohere up on in teracting with the en vironmen t. This en tanglemen t d estro ying pro cess limits the length of the computations that can b e realized by exp erimen t. Im- plemen ting qu an tum algorithms as circuits of lo w dep th can provi de a wa y to p er f orm as m u ch computation as p ossible within the limited time a v ailable, and for this r eason there is significan t in terest in fin d ing short quantum circuits for imp ortan t problems. Log-depth quantum circuits ha ve b een found for seve r al significant p roblems including the approxima te quan tum F ourier transform [3] and the en co d ing and deco d ing op erations for m any quantum error correcting cod es [10]. In addition to these applicatio ns, a pro- cedure for paralleliz in g to log-depth a large class of quan tum circuits has recen tly b een disco ve r ed [2]. These examples demonstrate th e surprising p o w er of short quan tu m circuits. Muc h of the w ork on quantum circuits is done in the standard mo del of u nitary quant um circuits on p ure states. In th is pap er a sligh tly differen t mo del of computation is considered: the mo del of m ixed s tate quant um circuits, in tro duced by Ah arono v, Kitaev, and Nisan [1]. While muc h of the previous complexity-t h eoretic wo r k on sh ort quan tu m circuits has b een in th e unitary mod el [4, 6], there h as also b een w ork outside of this mod el [13]. There are sev eral adv an tages to considering the more general mo d el of m ixed state circuits. The primary adv an tage is that the mixed state mo del is able to capture an y pro cess allo we d by quan tu m mechanics, so that results on this mo del ma y ha v e implications for exp erim ental w ork in quan tum computing. Th e pr oblem of distinguishing circuits m a y thus b e though t of as the problem of distin gu ish ing p oten tially n oisy ph ysical pro cesses. As an example, Key wor ds and phr ases: qu antum information, comput ational complexity , quantum circuits, qu antum intera ctive pro of sy stems. Submitted to ST ACS (Symposium on Theoretical Aspects o f Compu ter Science) 1 2 BILL ROSGEN finding an err or in an im p lemen tation of a quan tu m algorithm is simply the problem of distinguishing th e constructed circuit from one that is kno wn to b e correct. Unfortunately , in th is pap er it is sho wn that the apparent p ow er of short quan tu m computations comes w ith a price: log arith m ic depth quan tum circuits are e xactly as difficult to distinguish as p olynomial depth quan tum circuits. This equiv alence implies the sur p rising result that distinguish ing log-depth quantum computations is complete for the class QIP, the set of all problems that ha ve quant u m in teractiv e pro of systems. As PSP A CE ⊆ Q IP ⊆ EXP [8], this result also implies th at the p roblem is PS P A CE-hard . The result on circuit distin gu ish abilit y is sho wn usin g the closely related problem of determining if t wo circuits can b e made to output states that are close together. This problem wa s introdu ced b y Kitae v and W atrous [8 ] who sh ow it to b e b oth complete for QIP and con tained in EXP. The main result of th e present pap er is obtained by reducing an instance of this problem of p olynomial d ep th to an equiv alen t instance of logarithmic depth . This demonstrates that the problem of close images remains complete f or QIP eve n under a logarithmic depth restriction. Th e hardn ess of distinguishing sh ort quantum circu its is then demonstrated by a mo difi cation to the argumen t in [12] to sh o w th at the equiv alence of close images prob lem and the d istinguishabilit y holds even for log-depth circuits. The remainder of this pap er is organized as f ollo ws. In the next section, some of the notation and results that will b e needed are summarized. This is follo w ed by Sectio n 3, where the complete p roblems for QIP are discussed. In S ection 4 the r eduction from the p olynomial depth to logarithmic depth v ersions of the close images p roblem is giv en, and the correctness of th is construction is sho wn in Sectio n 5. Th e equiv alence b et ween the log-depth close images problem and the problem of distinguishin g log-depth computations is discussed in Section 6. 2. Preliminaries This s ection outlines some of the d efi nitions and results that will b e used throughout the pap er. F or a more thorough treatmen t of the concepts in tro du ced h ere see [9] and [11]. Throughout the pap er scripted lette r s suc h as H will refer to finite dim en sional Hilb ert spaces, D ( H ) will denote the set of all dens ity m atrices on H , and U ( H , K ) will denote the norm-pr eservin g linear op erators from H to K . The p ro of of the main result will mak e extensiv e use of t w o notions of distance b et w een quan tu m states. The first of these is the fidelit y . The fidelity bet w een t wo p ositiv e semidefinite op erators X and Y on a space H can b e defined as F ( X, Y ) = max {| h φ | ψ i | : | φ i , | ψ i ∈ H ⊗ K , tr K | φ ih φ | = X , tr K | ψ ih ψ | = Y } . This defin ition is kn o wn as Uhlmann’s Theorem, and it is used here as it is more directly applicable to the task at hand than the usual definition. As any purification of a state necessarily pur ifies the partial trace of that state, this equation implies th at the fidelit y is nondecreasing under the p artial trace. This prop ert y is kno wn as monot onicity and can b e stated more formally as F ( X, Y ) ≤ F (tr K X, tr K Y ) wh ere X , Y are p ositiv e semidefinite op erators on H ⊗ K . The fin al prop ert y of the fi d elit y that will b e n eeded is the result that the maximum fid elit y of any outputs of t wo transformations is m ultiplicativ e with r esp ect to the tensor pr o duct. This resu lt can b e found in [9] (see Problem 11.10 and apply the m u ltiplicativit y of the diamond norm w ith resp ect to the tensor pro duct). DISTINGUISHING SHOR T QUANTUM COMPUT A TIONS 3 Theorem 2.1 (Kitaev, Sh en, and Vya lyi [9]) . F or any c ompletely p ositive tr ansfor mations Φ 1 , Φ 2 , Ψ 1 , Ψ 2 on states in H max ρ,ξ ∈ D ( H⊗H ) F ((Φ 1 ⊗ Φ 2 )( ρ ) , (Ψ 1 ⊗ Ψ 2 )( ξ )) = Y i =1 , 2 max ρ,ξ ∈ D ( H ) F(Φ i ( ρ ) , Ψ i ( ξ )) The second notion of distance that will b e used is the tr ac e norm , whic h can b e defi ned for any lin ear op erator X b y k X k tr = tr √ X ∗ X , or equiv alen tly as the sum of the singular v alues of X . This qu an tit y is a norm, and s o in p articular it satisfies the triangle inequalit y . Similar to the fi delit y , the trace norm is monotone under the partial trace, though in this case the trace norm is non-increasing under this op eration. The p ro ofs that follo w will mak e essen tial use of the F uchs-v an de Graaf Inequalities [5] that relate the trace norm and the fidelit y . F or an y densit y op erators ρ and ξ on the same s p ace, these in equ alities are 1 − F( ρ, ξ ) ≤ 1 2 k ρ − ξ k tr ≤ p 1 − F( ρ, ξ ) . In addition to these measures on quan tum states, it w ill b e helpful to hav e a distance measure on quan tum transformations. One such measure is the diamond norm , which for a completely p ositiv e transformation Φ on den sit y op erators on H is giv en b y k Φ k ⋄ = sup ρ ∈ D ( H⊗H )   (Φ ⊗ I L ( H ) )( ρ )   tr . This n orm is essen tial when considering trans formations as it represents the distinguisha- bilit y of t w o tr an s formations w hen a reference system is tak en into accoun t. Th e sim p le supremum of the trace norm ov er all inputs to the c hannel is not stable under the addition of a reference system, and so th e d iamond norm is used in place of the simp ler on e. Mo r e prop erties and a more thorough defin ition of this n orm can b e found in [9 ]. The circuit mo del that will b e u sed in this p ap er is the mixed state mo del introd uced b y Aharono v, Kitaev, and Nisan [1]. Circuits in this mo d el are comp osed of qubits that are acted up on b y arbitrary trace pr eservin g and completely p ositiv e op erations. This mo del allo ws for non-unitary op erations, suc h as measuremen t or the in tro duction of an cillary qubits, to occur in the mid d le of the circuit. It is i mp ortant to note that this mo del captures any ph ysical p ro cess that quantum mec hanics allo ws, and so in p articular, an y computation that can b e done on mixed states with measuremen ts can b e r epresen ted in this mo del. F ortunately this mo d el is p olynomially equiv alen t to the standard mo d el of unitary quantum circuits (with ancilla ) follo we d by measurement, as sho w n in [1]. This w ill allo w us to consid er only circuits comp osed of unitary gates from some finite basis of one and tw o qubit gates with the additional op erations of in tro d ucing qub its in the | 0 i state and measuring in th e computational b asis. This restriction can b e strengthened, aga in with no loss of generalit y , to assume that all ancillary qubits are introd uced at th e start of the circuit and that all m easuremen ts are p erformed at the end. W e will often add to this circuit mo del one additional gate: the unb ou n ded f an-out gate. T his gat e, in constan t depth , applies a control led-not op eration from one q u bit to an arbitrary num b er of output qubits. It is not clear that this gate is a reasonable c hoice in a standard b asis of gates, and so it will b e clearly mark ed when this gate is allo wed int o the circuit mo del u nder consider ation. As an example of th e p o we r of this gate it can b e used to build a constant depth circuit for the app ro ximate quantum F ourier transform [7]. This gate is consid ered here f or the sole reason that if it is included in the standard set of gate s , the main result will also hold for constan t depth circuits. 4 BILL ROSGEN Measured t W H H | 0 i Figure 1: A circuit implemen ting the swap test. F or spaces H and K of the same d im en sion, we use W ∈ U ( H ⊗ K , H ⊗ K ) to r epresen t the op eration that sw aps the states in the t wo sp aces. As W is a p erm u tation matrix when expressed in th e computational basis, and th e p erm u tation that it encod es is comp osed ex- clusiv ely of transp ositions, th e sw ap op eration is b oth hermitian and un itary . F urthermore, W c an ea sily b e implemented in constan t depth, as all of the required transp ositions can b e p erformed in p arallel. Th is op erator is the essen tial comp onen t of the swap test , wh er e a con trolled W o p eration is used to determine how close t w o states are to eac h other. A circuit p erformin g the sw ap test is giv en in Figure 1, where the measurement is p erformed in the computational basis. Another w ay to view the s w ap test is as a pro jectiv e m easur emen t on to the sym metric and an tisymmetric subspaces. The pr o jectio ns in th is m easuremen t are giv en by ( I + W ) / 2 and ( I − W ) / 2. This formulati on of the sw ap test is equiv alen t to the circuit pr esen ted in Figure 1 . It is not imm ediately clear ho w a con trolled op eration on n qubits, su c h as the con trolled- sw ap op eration used in the swa p test, can b e p erformed in depth logarithmic in n . The straigh tforward imp lemen tation requires using one control qu bit to con trol eac h of the gates in the op eration. Ho wev er, Mo ore and Nilsson [10] giv e a simple constru ction that allo ws suc h an op eration to b e p erformed in log-depth. Prop osition 2.2 (Moore and Nilsson) . Any lo g dep th op er ation on n qubits c ontr ol le d by one qubit c an b e implemente d in O (log n ) depth with O ( n ) ancil lary qubits. Mo ore and Nilsson pro ve th is only for the constan t depth case, but the metho d of pr o of that they u se imm ediately extends to the log depth case. They pro ve this prop osition by using a tree of log n con trolled-not op er ations to ‘du p licate’ the control qub it. These copies can then b e us ed to control the remaining op erations, w ith eac h con trol qubit u sed at most a logarithmic num b er of times. This prop ositio n, as an example, imp lies that the swap test circuit on n qubits sho wn in Figure 1 can b e implemen ted in d epth O (log n ). If the fan-out gate is allo wed into th e standard basis of gates, then con trolled op erations can b e p erf ormed w ith only constant depth o v erh ead. A circuit that p erforms this can b e obtained by simp ly using one fan-out gat e to make n copies (in the computational basis) of the con trol qub it on to ancillary qu bits. T h ese ‘copies’ m a y then b e us ed to cont r ol eac h of the n op erations, with a final app lication of the fan-out gate to restore th e ancilla ry qubits to th e | 0 i state. As cont rolled op erations will b e the only place that the circuits constructed here exceed constant depth, this will allo w the pro of of the main resu lt for constan t depth circuits with fan-out. DISTINGUISHING SHOR T QUANTUM COMPUT A TIONS 5 3. Complete Problems for QI P The Close Image s p r oblem, defin ed and shown to b e complete for QI P in [8] can b e stated as follo ws. Close Images. F or c onstan ts 0 < b < a ≤ 1 , the input c onsists of quantum cir cuits Q 1 and Q 2 that implement tr ansformations fr om H to K . The pr omise pr oblem is to distinguish the two c ases: Y es: F( Q 1 ( ρ ) , Q 2 ( ξ )) ≥ a for some ρ, ξ ∈ D ( H ) , No: F( Q 1 ( ρ ) , Q 2 ( ξ )) ≤ b for al l ρ, ξ ∈ D ( H ) . This is simply the pr ob lem of determining if there are inpu ts to Q 1 and Q 2 that cause them to output states that are nearly th e same. It w ill b e helpful to abbreviate the name of this problem as C I a,b . A closely related problem is that of distingu ish ing t wo qu an tum circuits. T his problem w as in tro duced and sho wn complete for QIP in [12]. Quan t um C ircuit Distinguishabilit y . F or c onstants 0 ≤ b < a ≤ 2 , the input c onsists of quantum cir cuits Q 1 and Q 2 that implement tr ansforma tions fr om H to K . The pr omise pr oblem is to distinguish the two c ases: Y es: k Q 1 − Q 2 k ⋄ ≥ a , No: k Q 1 − Q 2 k ⋄ ≤ b . Less formally , this problem asks: is there an inpu t density matrix ρ on w h ic h the circuits Q 1 and Q 2 can b e made to act d ifferen tly? Th is problem will b e referred to as QC D a,b . It is our aim to pr o v e that th ese pr oblems remain complete for QIP when restricted to circuits Q 1 and Q 2 that are of dep th log arithmic in the n u m b er of input qubits. This will b e ac hieved in the case of p erfect sound ness error, i.e. a = 1 , 2 in the ab ov e problem definitions. Both of these pr oblem remain complete for QIP in this case. This restriction serv es only to mak e these problems easier, as distinguishing the t wo cases for a w eak er promise can only b e more difficult, so the results of this pap er will also imply the hardness of the m ore general p roblems. T he log-depth versions of these problems will b e referr ed to as L og-depth CI 1 ,b and Log-d epth QCD 2 ,b , and since these are restrictions of Q IP-complete problems it is clear that they are also in QIP . Similarly , th e abbreviations Co nst-depth CI 1 ,b and Co nst-depth QCD 2 ,b for th e ve r sions of these problems on constan t-depth circuits will b e conv enien t. 4. Log-Depth Construction In this section the reduction from the general CI 1 ,b problem to th e log-depth restriction of the problem is describ ed. The general idea b ehind the construction is to simply slice the circuits of an instance of CI 1 ,b in to logarithmic-depth pieces and r un them in parallel. These circu its will require more input, bu t if eac h piece of the circuit is given as inpu t the same state output by the previous p iece, then the output of the last piece of the circuit will b e equal to the output of th e original circuit. This ma y n ot b e the case if the in termediate inputs are not the outputs of the previous pieces, and so additional tests that ensure these inputs are at least close to the desired states are required . T o describ e the reduction, let Q 1 and Q 2 b e the circuits from an instance of CI 1 ,b , and let n b e the size (n u mb er of gat es) of Q 1 and Q 2 (b y padding the smaller circuit, if necessary). In ord er to p erform the slicing of the circuit in to pieces it is assumed that Q 1 6 BILL ROSGEN Q ( ρ ) U 2 U n ρ · · · U 1 Figure 2: The original circuits Q 1 and Q 2 decomp osed int o constan t d epth u nitary circuits. and Q 2 first intro d uce an y necessary ancillary qub its, then apply lo cal u nitary gates, an d finally trace out an y qubits that are n ot part of the input. This restrictio n can b e made with no loss in generalit y , as an y quan tum circuit, eve n one that in corp orates measurements and other non-unitary op erations, can b e app ro ximated b y suc h a circuit, an d furtherm ore, this circuit uses a num b er of gate s that is a p olynomial in the size of th e original circuit [1]. A simple w ay to decomp ose Q 1 in to constan t depth pieces is to simply let eac h gate of Q 1 b e a p iece in the decomposition. Let U 1 , U 2 , . . . , U n b e these p ieces, with the additional complicatio n that the op eration U 1 b oth adds th e ancillary qubits and p erform s the first gate of the circuit. In a similar wa y , Q 2 can b e d ecomp osed in to constan t depth pieces V 1 , V 2 , . . . , V n . These pieces are sho wn in Figure 2. If Q 1 and Q 2 implemen t transformations from H to K , usin g ancillary qub its that fi t in to A , and trace out the qubits in B , then the spaces H ⊗ A and B ⊗ K are isomorphic, since b y assumption Q 1 and Q 2 first intro d uce an y n eeded ancilla and only trace qub its out at the end of the computation. Using these spaces, and implicitly this isomorphism, we ha ve U 1 , V 1 ∈ U ( H 1 , B 1 ⊗ K 1 ) U i , V i ∈ U ( H i ⊗ A i , B i ⊗ K i ) for 2 ≤ i ≤ n, where the subscripted sp aces are copies of the non-sub s cripted sp aces that hold the input or output of one of the p ieces of the original circuits. As an example of this notatio n, if ρ ∈ D ( H ), then the output of Q 1 on ρ is giv en by tr B n U n U n − 1 · · · U 1 ρU ∗ 1 U ∗ 2 · · · U ∗ n , and the outp ut of Q 2 is give n by the same expression u s ing the V i op erators. Using th is decomp osition of Q 1 and Q 2 , circuits C 1 and C 2 are constru cted that are logarithmic in d epth and still in some sense faithfu lly imp lemen t Q 1 and Q 2 . This is done b y runn ing the circuits corresp onding to U 1 , . . . , U n in parallel, and tracing out all the qubits that are not in the output of U n . S uc h a circuit is constant d epth, b ut do es not necessarily output a state in the image of Q 1 , as the in put to U i is not necessarily close to the output f r om U i − 1 . This problem can b e dealt with by comparing the ou tp ut of U i − 1 to th e in put to U i . I n ord er to do this in logarithmic dep th an auxiliary inpu t that is first compared against the inpu t to U i and then held in reserve to compare to the output of U i − 1 is needed. T o compare th ese qu an tum states the swap test can b e u sed. This test will fail with some pr obabilit y d ep ending on the distance b et w een the tw o stat es. An example of the construction used to ensu re that the output of U i − 1 agrees with the input to U i is giv en in Figure 3. T o simp lify the analysis of the constru cted circuits these tests are con trolled so that either one or the other is p erformed. This will affect th e failure p robabilit y b y a factor of at most tw o, but will allo w the analysis of eac h sw ap test to ignore the effect of th e DISTINGUISHING SHOR T QUANTUM COMPUT A TIONS 7 | ψ i i t t U i swap test U i +1 swap test X H | 0 i | ψ i +2 i | ψ i +1 i | ψ i +1 i t Figure 3: T esting that the outpu t of U i is close to the inp ut of U i +1 . T h e in puts | ψ j i are the ideal inputs to U j , and are lab elled for clarit y only – no assu m ptions are m ade ab out these states. Qubits that do not reac h the righ t edge are traced out. other. T o imp lemen t this a con trol qubit is u sed so that either the first or the second test is p erformed b etw een ev ery tw o p ieces U i , U i +1 of the circuit. I f a test is not p erformed, then the v alue of the output qub it of the swa p test is left unc h anged, and so the result of the test is a qubit in the | 0 i state. Th ese con tr olled op erations can b e implemente d in logarithmic depth u sing the tec hn ique of Mo ore and Nilsson [10]. After adding these tests b et w een eac h piece of the circuit there is one fi nal mo dification required. If any of the sw ap tests fail, i.e. detect states that are not the s ame, then they will output qubits in the | 1 i stat e. As y es instances of CI 1 ,b ha ve outputs that are close together, we can ensur e that no ou tp uts of th e constructed circuits can b e close if an y sw ap tests fail b y adding dum m y qubits in the | 0 i stat e to b e compared to the outputs of the sw ap tests in the other circuit. These dummy qu b its are sh o wn in Figure 4. The constructed circuits C 1 and C 2 are obtained b y decomp osing Q 1 and Q 2 in to constan t depth pieces, inserting the swa p tests sho w n in Figure 3, and adding d ummy qubits to ensure that the sw ap tests in the other circuit do not fail. At the end of these circuits, all qubits are traced out, except the output (in the space K n ) of U n or V n , the output of the sw ap tests, and the dummy zero qubits. If the outputs of C 1 and C 2 are close together, then intuitiv ely the outpu t of the swa p tests in eac h circuit m ust b e close to zero and the output of U n and V n m u s t also b e close. If the sw ap tests d o not fail with high probabilit y (i.e. the outputs are close to zero), then these circuits will m ore or less faithfully repro du ce the outp ut of Q 1 and Q 2 . Thus, in the case that the outputs of C 1 and C 2 can b e m ade close, w e will b e able to argue th at the output of Q 1 and Q 2 can also be made close. Pro ving th at this intuitiv e picture is accurate forms the con tent of the next section. In the other direction, it is not hard to see that if there are states ρ, ξ ∈ D ( H ) such that Q 1 ( ρ ) = Q 2 ( ξ ), then there are similar states for the constructed circuits C 1 and C 2 . T o do this, notice that the circuit construction do es n ot change if additional qu bits are added to the circu its to allo w purification of the states ρ and ξ to b e used as inputs to C 1 and C 2 . These additional qub its are traced out with the other qu bits at the end of the circuit, so that the output state of the circuit are not changed. As these p urifications are pur e states and all op erations p erformed durin g the circuit are unitary , th e int er m ediate states of the circuits must also b e pure states. If the inp ut state to C 1 is | ψ i , then b y pr o viding the state | ψ i ⊗ U 1 | ψ i ⊗ · · · ⊗ U n − 1 U n − 2 · · · U 1 | ψ i 8 BILL ROSGEN Output of U n C 2 C 1 . . . . . . . . . . . . . . . . . . ) | 0 i ⊗ n ) Sw ap tests ) Sw ap tests ) | 0 i ⊗ n Output of V n Figure 4: The outputs of C 1 and C 2 . as input to C 1 , the outpu t of eac h block of the circuit will b e iden tical to th e inpu t to the next blo c k, ensuring that all the swa p tests will su cceed with probability one. It remains only to c hec k on such inp ut s tates that C 1 pro du ces the same output as Q 1 on ρ . This can b e observed b y noting that the outpu t of the circuit is exactly tr B n U n U n − 1 · · · U 1 ρU ∗ 1 U ∗ 2 · · · U ∗ n , whic h b y construction is equal to the outp u t of Q 1 on ρ . Th us if the circuits Q 1 and Q 2 ha ve intersec ting images then so do the circuits C 1 and C 2 . Th is observ ation pro v es the completeness of the construction. Sound ness is consider ab ly more in tricate, and is the fo cus of the n ext section. 5. Soundness of t he Construction In this section it is demonstrated th at if the images of the original circuits Q 1 and Q 2 are far apart then so must b e the images of the constructed circuits C 1 and C 2 . As the constructed circuits essen tially sim ulate Q 1 and Q 2 the desired result can b e obtained b y arguing that either the outputs of C 1 and C 2 are far apart or the input to at least one of the constructed circuits is not a faithful sim ulation of the corresp ondin g original circuit. In the case that th is sim u lation is not f aithful it w ill b e sh own that there is some sw ap test that fails with reasonable pr obabilit y . Th is implies that outputs of the constru cted circu its m ust also b e distan t, as the f ailing swa p test pro duces a state of the form (1 − p ) | 0 ih 0 | + p | 1 ih 1 | that has lo w fidelit y with the corresp ond ing dummy zero qubit of the other circuit. As a fi rst s tep, w e place a lo we r b ound on th e f ailure probabilit y of a swap test in terms of the fid elity of the t w o stat es b eing compared. In the follo wing lemma the sw ap test is view ed as a measurement of the symmetric and an tisymm etric pro jectors, w ith the outcome that pr o duces a qub it in the state | 1 i corresp onding to the an tisymmetric case. Lemma 5.1. If ρ ∈ D ( A ⊗ B ) then a swap test on A ⊗ B r eturns the antisymmetric out- c ome with pr ob ability at le ast 1 2 − 1 2 F(tr A ρ, tr B ρ ) . Pr o of. Let | ψ i ∈ A ⊗ B ⊗ C b e a purifi cation of ρ , where C is an arb itrary space of su fficien t dimension to allo w su ch a purification. The swap test measur es the state on A ⊗ B with the p ro jectors 1 2 ( I − W ) and 1 2 ( I + W ), w h ere W is the swap op erator on A ⊗ B . Thus, the DISTINGUISHING SHOR T QUANTUM COMPUT A TIONS 9 an tisymmetric outcome o ccur s with pr obabilit y 1 4 tr ([( I − W ) ⊗ I ] | ψ ih ψ | [( I − W ∗ ) ⊗ I ]) = 1 2 h ψ | I ⊗ I − W ⊗ I | ψ i = 1 2 − 1 2 h ψ | W ⊗ I | ψ i , as W is hermitian. Then as W is also un itary , the states | ψ i and W | ψ i eac h purify b oth tr A⊗C | ψ ih ψ | and tr B⊗C | ψ ih ψ | , and so b y Uhlmann’s theorem 1 2 − 1 2 h ψ | W ⊗ I | ψ i ≥ 1 2 − 1 2 F(tr A⊗C | ψ ih ψ | , tr B⊗C | ψ ih ψ | ) . After tracing out the s p ace C , th is is exactly the statemen t of the lemma. This lemma cannot b e immediately applied to the circuits C 1 and C 2 , as in these circuits the output of one blo c k of the circuit is not directly compared to the input to the next blo c k, but in stead eac h of these states are with probabilit y 1 / 2 compared to some intermediate v alue. In order to deal with this difficult y , w e use the F uchs-v an de Graaf inequalities to translate the fidelit y to a relation in vo lvin g the trace norm, whic h w e can then apply the triangle inequalit y to. T h is application of the tr iangle inequalit y shows that at lea st one of the tw o s w ap tests fails with probabilit y b ounded b elo w by an expression in v olving the fidelit y . In the f ollo wing corollary the redu ced states of v arious p arts of the input to either of the circuits C 1 or C 2 are used, bu t it is not assumed that these s tates are giv en in a separable form. F or instance, th e d ensit y matrices ρ i , σ i , and ξ i that app ear in the lemma ma y b e part of some larger entangle d pure state , so th at the failure p robabilities of the tw o sw ap tests need not b e indep endent. Corollary 5.2. If | ψ i is input to the cir cuit C a for a ∈ { 1 , 2 } , with ρ i the r e duc e d state of | ψ ih ψ | on H i ⊗ A i , then at le ast one of the swap tests on the i th blo c k of C a fails with pr ob ability at le ast 1 64 k U i ρ i − 1 U ∗ i − ρ i k 2 tr . Pr o of. In the i th blo c k of C a there are t w o inputs to the first swa p test: let the r ed uced densit y op erators of these inputs b e ρ i and σ i . Th e inp uts to the second sw ap test are then giv en by σ i and U i ρ i − 1 U ∗ i = ξ i . As exactly one of these tests is p erformed we d o n ot n eed to consider the effect of the first test on th e state when considering the second test, and so the same in put state σ i is used in b oth sw ap tests. By Lemma 5.1, the failure probabilit y of first and seco n d tests, when p erformed, are at least 1 2 (1 − F( ρ i , σ i )) and 1 2 (1 − F( σ i , ξ i )), resp ectiv ely . Thus, the p robabilit y p that at least one of these tests fails, giv en that eac h of th em is p erformed with p robabilit y 1 / 2 , is at least p ≥ 1 2 max  1 2 (1 − F( σ i , ξ i )) , 1 2 (1 − F( ρ i , σ i ))  = 1 4 (1 − min { F( σ i , ξ i )) , F ( ρ i , σ i ) } ) . By th e F uc hs-v an de Gr aaf inequalities, this fidelit y ma y b e replace d b y the trace norm. Doing so, we obtain p ≥ 1 16 max( k σ i − ξ i k 2 tr , k ρ i − σ i k 2 tr ) . Finally , as this maximum m u st b e at least the a v er age of the t wo v alues, p ≥ 1 16  k σ i − ξ i k tr 2 + k ρ i − σ i k tr 2  2 ≥ 1 64 k ρ i − ξ i k 2 tr , where the last inequalit y follo w s from an application of th e triangle inequ ality . 10 BILL ROSGEN By rep eatedly app lyin g some of the prop erties of the trace norm discussed in S ection 2 it is somewhat tedious but not difficult to reduce the problem at hand to the previous Corollary . This is the conten t of the follo wing theorem. Theorem 5.3. If F( Q 1 ( ρ 0 ) , Q 2 ( ξ 0 )) < 1 − c for al l ρ 0 , ξ 0 ∈ H then F( C 1 ( ρ ) , C 2 ( ξ )) < 1 − c 2 144 n 2 for al l ρ, ξ ∈ ( H ⊗ A ) ⊗ 2 n . Pr o of. Let ρ and ξ b e inputs to C 1 and C 2 , and let ρ i , ξ i b e the reduced s tates of these inputs on H i ⊗ A i for 0 ≤ i ≤ 2 n , where the states for i > n are th e inpu ts th at are only used by th e swa p tests, whic h we will not need to refer to explicitly . That is, ρ i and ξ i for 0 ≤ i ≤ n are the p ortions of the state that are inpu t to the un itaries U i and V i that make up th e circuits Q 1 and Q 2 . The outp ut of the circuits C 1 and C 2 is th en giv en by a num b er of qubits corresp ond ing to the sw ap tests as well as the states tr B n ρ n and tr B n ξ n , where B n is simply the space that is traced out to obtain the outpu t from the unitary represen tations of the original circuits. By the condition on the fidelity of Q 1 and Q 2 and the F uc hs-v an de Graaf inequalities, w e ha ve 2 c < k Q 1 ( ρ 0 ) − Q 2 ( ξ 0 ) k tr . Using th e triangle inequalit y we can relate this to the distance b et w een the constructed circuits. Adding terms and simplifying, w e obtain 2 c < k Q 1 ( ρ 0 ) − tr B n ρ n + tr B n ξ n − Q 2 ( ξ 0 ) + tr B n ρ n − tr B n ξ n k tr ≤ k Q 1 ( ρ 0 ) − tr B n ρ n k tr + k tr B n ξ n − Q 2 ( ξ 0 ) k tr + k tr B n ρ n − tr B n ξ n k tr . W e no w observ e that k tr B n ρ n − tr B n ξ n k tr ≤ k C 1 ( ρ ) − C 2 ( ξ ) k tr b y the monotonicit y of the trace norm u n der the partial trace, s in ce th e f orm er can b e obtained from the later b y tracing out the appr opriate spaces. Using this we ha ve 2 c < k Q 1 ( ρ 0 ) − tr B n ρ n k tr + k tr B n ξ n − Q 2 ( ξ 0 ) k tr + k C 1 ( ρ ) − C 2 ( ξ ) k tr (5.1) As the three terms on the righ t are nonn egativ e, at least one of them must b e larger than the a v erage 2 c/ 3. If k C 1 ( ρ ) − C 2 ( ξ ) k tr > 2 c/ 3 then F( C 1 ( ρ ) , C 2 ( ξ )) < 1 − c 2 / 144 and there is nothing left to p ro ve. The cases where one of the first t w o terms of (5.1) excee ds 2 c/ 3 are symmetric, and so w e can consid er only the first term. Expanding Q 1 ( ρ 0 ) in terms of the U i , we obtain 2 c 3 < k Q 1 ( ρ 0 ) − tr B n ρ n k tr = k tr B n U n U n − 1 · · · U 1 ρ 0 U ∗ 1 U ∗ 2 · · · U ∗ n − tr B n ρ n k tr ≤ k U n U n − 1 · · · U 1 ρ 0 U ∗ 1 U ∗ 2 · · · U ∗ n − ρ n k tr , where once again the monotonicit y of the trace norm u nder the partial trace has b een used . By rep eating th e strategy of add in g terms and then applying the triangle inequalit y w e ha ve 2 c 3 < k U 1 ρ 0 U ∗ 1 − ρ 1 k tr + k U n U n − 1 · · · U 2 ρ 1 U ∗ 2 U ∗ 3 · · · U ∗ n − ρ n k tr . Here we ha ve made use of th e unitary inv ariance of the trace norm to discard the op erators U 2 , . . . U n from the first term. Con tinuing in this fashion we ha ve 2 c 3 < n X i =1 k U i ρ i − 1 U ∗ i − ρ i k tr . DISTINGUISHING SHOR T QUANTUM COMPUT A TIONS 11 As all terms in this sum are n onnegativ e, there must b e at least one term in the sum that exceeds 2 c/ (3 n ), as this is a lo wer b ound on the a v erage of all terms. Th us, for some v alue of i , we ha ve k U i ρ i − 1 U ∗ i − ρ i k tr > 2 c/ (3 n ) , and so by Corollary 5.2 one of th e corresp on d ing sw ap tests fails with prob ab ility p > c 2 / (144 n 2 ). T h e qubit representing the outpu t v alue of this swap test is then of the form (1 − p ) | 0 ih 0 | + p | 1 ih 1 | , and so, b y the monotonicit y of the fi d elit y under the p artial trace, F( C 1 ( ρ ) , C 2 ( ξ )) ≤ F((1 − p ) | 0 ih 0 | + p | 1 ih 1 | , | 0 ih 0 | ) = 1 − p < 1 − c 2 144 n 2 , as in the statemen t of the theorem. By com b in ing T heorem 5.3 with the obs erv atio n in Section 4 and the m u ltiplicativit y of the m axim um output fid elit y of t wo transf orm ations, w e obtain the follo wing result. Corollary 5.4. The pr oblem L og-depth CI 1 ,b is QIP -c omplete for any c onsta nt 0 < b < 1 . Pr o of. Theorem 5.3 establishes the completeness of the problem for an y b ≥ 1 − c 2 / (144 n 2 ), where n is an upp er b ound on the size of the circu its. Using Theorem 2.1 of K itaev, Shen, and Vy alyi [9] w e can rep eat eac h of the circuits r times in parallel to obtain the completeness of the problem for b ≥  1 − c 2 / (144 n 2 )  r , which can b e mad e smaller than an y constan t for r some p olynomial in n . As the circuits constructed by the redu ction only mak e use of logarithmic depth wh en p erformin g swap tests, and the con trolled sw ap op erations p erformed by these tests can b e accomplished in constan t depth using u n b ounded fan-out gates, th e follo wing C orollary follo ws immediately f r om the p revious one. Corollary 5.5. The pr oblem Const-depth CI 1 ,b on cir cuits with the unb ounde d fan-out gate is QI P -c omplete for for any c onsta nt 0 < b < 1 . 6. Distinguishing Log-Depth C omputations The hardness of Log-depth CI 1 ,b can b e extended to L og-depth QCD 2 ,b b y observin g that the reduction for the p olynomial dep th version of this problem in [12] can b e made to preserve the depth of the constructed circuits. Once th is obser v ation is made, the hardness of the log-depth (and constan t-depth with fan-out) versions of the circuit distinguishabilit y problem is immediate. The reduction in [12] tak es as input circuits ( Q 1 , Q 2 ) and pr o duces circuits C 1 and C 2 . Without describin g the reduction in detail, th e constructed circuits C 1 and C 2 run, dep endin g on the v alue of a cont rol qubit, one of Q 1 and Q 2 , follo we d by a constan t depth circuit. If the input circuits Q 1 and Q 2 ha ve logarithmic depth, then the only significan t difficult y is the fact that con trolled v ersions of these circuits are needed. Ho w ev er, as we ha ve already seen, if w e replace the gates in Q 1 and Q 2 with control led ve rsions, then w e can use the scheme of Mo ore and Nilsson [10] to implement the con trolled op erations in logarithmic depth. With this mo dification, the r eduction in [12] can b e reused to sho w the hardness of the QCD pr oblem on log-depth circuits. Corollary 6.1. L og-depth QCD 2 ,b is QI P -c omplete for any c onstant 0 < b < 2 . Once ag ain these con trolled op erations can b e implemented in a constan t depth circuit if the un b ound ed f an-out gate is allo w ed in to th e set of allo w ed gate s . 12 BILL ROSGEN Corollary 6.2. Const-depth QCD 2 ,b on cir cuits with the unb ounde d fan-out gate is Q I P - c omplete for any c onstant 0 < b < 2 . 7. Conclusion The hardn ess of distinguishing eve n log-depth mixed state quant u m circuits leav es sev eral related op en problems, a f ew of w hic h are listed here. • Can this n ew complete p roblem b e us ed to fur ther understand QIP? • Does this result rely in an essentia l w ay on the mixed state circuit m o del? Ho w difficult is it to distinguish quan tum circuits in less general mo dels of computation? • What is the complexit y of distinguish ing constan t depth quantum circuits that d o not use the unb ou n ded fan-out gate? 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