Eliminating Intermediate Measurements in Space-Bounded Quantum Computation
aa r X i v : . [ c s . CC ] J un Eliminating Intermediate Measurements in Space-BoundedQuantum Computation
Bill FeffermanDepartment of Computer Science, The University of ChicagoZachary Remscrim ∗ Department of Mathematics, MIT
Abstract
A foundational result in the theory of quantum computation known as the “principle of safestorage” shows that it is always possible to take a quantum circuit and produce an equivalentcircuit that makes all measurements at the end of the computation. While this procedure istime efficient, meaning that it does not introduce a large overhead in the number of gates, it usesextra ancillary qubits and so is not generally space efficient. It is quite natural to ask whetherit is possible to defer measurements to the end of a quantum computation without increasingthe number of ancillary qubits.We give an affirmative answer to this question by exhibiting a procedure to eliminate allintermediate measurements that is simultaneously space-efficient and time-efficient. A key com-ponent of our approach, which may be of independent interest, involves showing that the well-conditioned versions of many standard linear-algebraic problems may be solved by a quantumcomputer in less space than seems possible by a classical computer.
Quantum computation has the potential to obtain dramatic speedups for important problems suchas quantum simulation (see, e.g., [19, 28]) and integer factorization [43]. While fully scalable, fault-tolerant quantum computers may still be far from fruition, we have now entered at an exciting erain which impressive but resource constrained quantum experiments are being implemented in manyacademic and industrial labs. As the field transitions from “proof of principle” demonstrations ofprovable quantum advantage to solving useful problems on near-term experiments, it is particularlycritical to characterize the algorithmic power of feasible models of quantum computations that haverestrictive resources such as “time” (i.e., the number of gates in the circuit) and “space” (i.e., thenumber of qubits on which the circuit operates) and to understand how these resources can betraded-off.A foundational question in this area asks if it is possible to space-efficiently defer intermediatemeasurements in a quantum computation (see e.g., [18, 23, 31, 35, 45, 49–51]). While a classic resultknown as the “principle of safe storage” states that it is always possible to time-efficiently defer ∗ Corresponding author, [email protected] Q acts on s qubits and performs m intermediate measurements, the circuit Q ′ constructed using thisprinciple operates on s + poly ( m ) qubits; if, for example, s = O (log t ) and m = Θ( t ), this entailsan exponential blowup in the amount of needed space .Our main result solves this problem. We show that every problem solvable with a quantumcomputation that acts on a designated number of qubits can also be solved by a “unitary” quan-tum algorithm that uses the same space and makes all measurements at the end of the computa-tion. Stated more formally in the language of complexity theory, we let BQ U SPACE ( s ( n )) (resp. BQSPACE ( s ( n ))) denote the class of promise problems recognizable with two-sided bounded-errorby a uniform family of unitary (resp. general) quantum circuits, where, for each input of length n , there is a corresponding circuit that operates on O ( s ( n )) qubits and has 2 O ( s ( n )) gates. Notethat it is standard to require that the running time of a computation is at most exponential in itsspace-bound; see, for instance, [31, 49, 51] for the importance of this restriction in quantum space-bounded computation, as well as [39] for the importance of the analogous restriction in probabilisticspace-bounded computation. Furthermore, let QMASPACE ( s ( n )) denote those promise problemsrecognized by a quantum Merlin-Arthur protocol that operates in space O ( s ( n )) and time 2 O ( s ( n )) .Our main result is: Theorem 1.
For any space-constructible function s : N → N , where s ( n ) = Ω(log n ) , we have BQ U SPACE ( s ( n )) = BQSPACE ( s ( n )) = QMASPACE ( s ( n )) . Remark.
To the best of our knowledge, the containment
BPSPACE ( s ( n )) ⊆ BQ U SPACE ( s ( n )) wasnot known to hold, where BPSPACE ( s ( n )) denotes the analogously defined class of language recog-nizable by a probabilistic algorithm in space O ( s ( n )) (and time 2 O ( s ( n )) ); see, for instance, [31, 50]for previous discussion of this question. As one would expect quantum computation to gener-alize probabilistic computation, the lack of a proof of this containment was unfortunate. Sinceit is clear that BPSPACE ( s ( n )) ⊆ BQSPACE ( s ( n )), we have as a corollary of Theorem 1, that BPSPACE ( s ( n )) ⊆ BQ U SPACE ( s ( n )), resolving this question.Before proceeding further, it is worthwhile to briefly discuss why it is desirable to be able toeliminate intermediate measurements. Firstly, quantum measurements are a natural resource, muchas time and space are; in addition to the general desirability of using as few resources as possible inany sort of computational task, it is especially desirable to avoid intermediate measurements, dueto the technical challenges involved in implementing such measurements and resetting such qubitsto their initial states (for a discussion of these issues from an experimental perspective see, e.g.,[14]). Secondly, unitary computations are reversible . The ability to “undo” a unitary subroutine,by running that subroutine in reverse, is used routinely in the design and analysis of quantumalgorithms (see, for instance, [6, 17, 18, 30, 32, 44, 53]). Moreover, reversible computations may beperformed without generating heat [27]. Thirdly, by demonstrating that unitary quantum spaceand general quantum space are (polynomially) equivalent in power, we show that the definition ofquantum space is quite robust , as allowing intermediate measurements, or even general quantumoperations, does not affect the definition of BQ U SPACE ( s ( n )).We also study the one-sided (bounded-error and unbounded-error) analogues of the aforemen-tioned two-sided bounded-error space-bounded quantum complexity classes. We show the followingresults (see Section 2.2 for definitions of the complexity classes appearing in the following theorems). Theorem 2.
For any space-constructible function s : N → N , where s ( n ) = Ω(log n ) , we have RQMASPACE ( s ( n )) = RQ U SPACE ( s ( n )) ⊆ RQSPACE ( s ( n )) ⊆ coQMASPACE ( s ( n )) . heorem 3. For any space-constructible function s : N → N , where s ( n ) = Ω(log n ) , we have NQMASPACE ( s ( n )) = NQ U SPACE ( s ( n )) = NQSPACE ( s ( n ))= coPreciseQMA SPACE ( s ( n )) = coC = SPACE ( s ( n )) . In order to prove the theorems stated above, we study the complexity of solving various standardlinear-algebraic problems, both exactly and approximately. Let intDET denote the problem ofcomputing the determinant of an n × n integer-valued matrix, and, following its original definition byCook [12], let DET ∗ denote the class of problems NC (Turing) reducible to intDET (the definition ofthis class is somewhat delicate, see [4,5,29] for further discussion). Let BQ U L = BQ U SPACE (log( n )), BQL = BQSPACE (log( n )), and BPL = BPSPACE (log( n )) denote the bounded-error quantum andprobabilistic logspace classes. Finally, let PQ U L , PQL , and PL denote the corresponding unbounded-error classes. Before our work, the following relationships were known [12, 49, 51]: NC ⊆ L ⊆ RL ⊆ BPL ⊆ BQL ⊆ PL = PQ U L = PQL ⊆ DET ∗ ⊆ NC ⊆ L , RL ⊆ NL ⊆ PL , and L ⊆ BQ U L ⊆ BQL . Many natural linear-algebraic problems are complete for
DET ∗ , including intDET , intMATINV (the problem of computing a single entry of the inverse of an invertible integer-valued matrix), and intMATPOW (the problem of computing a single entry of the m th power of an n × n integer-valuedmatrix, for m = poly ( n )). It seems rather unlikely that any such DET ∗ -complete problem is inthe class BQL , as this would imply
BQL = DET ∗ , and, therefore, NL ⊆ BQL . We next considerthe problem of approximating the answer to these, and other related, linear-algebraic problems. Inparticular, let poly -conditioned-
MATINV denote the promise problem of approximating (to additive1 /poly ( n ) accuracy) a single entry of the inverse of an n × n matrix A with condition number κ ( A ) = poly ( n ) (see Section 3 for a precise definition of this problem). Ta-Shma [45], building onthe landmark result of Harrow, Hassidim, and Lloyd [20], showed poly -conditioned- MATINV ∈ BQL ;Fefferman and Lin [18] improved upon this result by showing that poly -conditioned-
MATINV is, infact, BQ U L -complete, thereby exhibiting the first known natural BQ U L -complete (promise) problem.In Section 3.1, we extend this result by showing that the poly -conditioned versions of many standard DET ∗ -complete problems are BQ U L -complete. Theorem 4.
Each poly -conditioned promise problem given in Definition 15 is BQ U L -complete. This is an interesting result, in and of itself, as it demonstrates an intriguing relationshipbetween BQ U L and DET ∗ . Moreover, this result will then allow us to prove Theorem 1 and theother main results of this paper. In particular, in Section 3.2, we will show that the problem poly -conditioned- ITMATPROD is BQL -hard (again, see Section 3 for a precise definition of thisproblem). By the preceding theorem, poly -conditioned-
ITMATPROD ∈ BQ U L , which implies BQL = BQ U L ; Theorem 1, which states the more general equivalence for any larger (space-constructible)space bound, then follows from a standard padding argument. In Section 3.3, we continue the studyof fully logarithmic approximation schemes , initiated by Doron and Ta-Shma [16], and show thatthe BQL vs.
BPL question is equivalent to several distinct questions involving the relative power ofquantum and probabilistic fully logarithmic approximation schemes.Consider κ ( n )-conditioned- DET , the problem of approximating, to within a multiplicative factor(1 + 1 /poly ( n )), the determinant of an n × n with condition number κ ( n ). Boix-Adser`a, Eldar, and3ehraban [8] recently showed κ ( n )-conditioned- DET ∈ DSPACE (log( n ) log( κ ( n )) poly (log log n )).Furthermore, they raised the following question: is poly -conditioned- DET BQL -complete? As animmediate consequence of Theorem 4, we answer their question in the affirmative.
Proposition 5. poly -conditioned-
DET is BQ U L -complete (and, therefore, BQL -complete)
To briefly explain the significance of the question posed by Boix-Adser`a, Eldar, and Mehra-ban [8], note that their result shows poly -conditioned-
DET ∈ DSPACE (log ( n ) poly (log log n )).Therefore, as poly -conditioned- DET is BQL -complete, the statement
BQL ⊆ DSPACE (log − ǫ n )would follow from either a small improvement in their result (i.e., proving a stronger upper boundon the needed amount of deterministic space in terms of κ ( n )) or from a small improvement inour result (i.e., proving κ ( n )-conditioned- DET remains
BQL -hard for subpolynomial κ ( n )). Recallthat the strongest currently known “dequantumization” result of this type is the classic result ofWatrous [51], which states BQL ⊆ DSPACE (log n ) (cf. the strongest currently known “derandom-ization” result of this type, given by Saks and Zhou [40], which states BPL ⊆ DSPACE (log n )). Inparticular, if BQL DSPACE (log − ǫ n ), ∀ ǫ >
0, then both our result and the result of Boix-Adser`a,Eldar, and Mehraban are essentially optimal.In Section 4, we study well-conditioned versions of the “standard” C = L -complete problems.We will observe that the relationship of these problems to the one -sided error versions of space-bounded quantum complexity classes is very much analogous to the relationship, discussed earlier,of the well-conditioned versions of the “standard” DET ∗ -complete problems to the two -sided errorversions of space-bounded quantum complexity classes. This enables us to prove Theorem 2 andTheorem 3, which concern the relative power of unitary and general quantum space in the one-sidederror cases.In Section 5, we establish the BQ U L -completeness of “scaled-down” versions of certain QMA -complete problems and certain
DQC1 -complete problems. We conclude by stating several openproblems related to our work in Section 6.
Let Mat( n ) = C n × n denote the set of all n × n complex matrices and let Herm( n ) = { A ∈ Mat( n ) : A = A † } denote the set of all n × n Hermitian matrices. For A ∈ Mat( n ), let σ ( A ) ≥ · · · ≥ σ n ( A ) ≥ λ ( A ) , . . . , λ n ( A ) ∈ C denote its eigenvalues; if A ∈ Herm( n ),then λ j ( A ) ∈ R , ∀ j , and we order the eigenvalues such that λ ( A ) ≥ · · · ≥ λ n ( A ). Let I n denote the n × n identity matrix, Pos( n ) = { A ∈ Herm( n ) : λ n ( A ) ≥ } denote the n × n positive semidefinitematrices, Proj( n ) = { A ∈ Pos( n ) : A = A } denote the n × n projection matrices, U( n ) = { A ∈ Mat( n ) : AA † = I n } denote the n × n unitary matrices, and Den( n ) = { A ∈ Pos( V ) : tr( A ) = 1 } denote the n × n density matrices. Let Q [ i ] n = { r + cid : r, c, d ∈ Z , | r | , | c | , | d | ≤ O ( n ) } denote theset of all O ( n )-bit Gaussian rationals and let d Mat( n ) (resp. \ Herm( n ) , d Pos( n ) , etc.) denote thesubset of Mat( n ) (resp. Herm( n ) , Pos( n ) , etc.) which consist of those matrices whose entries are all O ( n )-bit Gaussian rationals. For a sequence of matrices A , . . . , A m , and for indices j , j , where1 ≤ j ≤ j ≤ m , let A j ,j = j Q j = j A j .We consider parameterized promise problems of the form P = ( P n,f ,...,f h ) n ∈ N , for functions f , . . . , f h : N → R ; P n,f ,...,f h consists of instances of size n which satisfy various conditions ex-pressed in terms of f ( n ) , . . . , f h ( n ). For a promise problem P defined over some alphabet Σ, we,4y slight abuse of notation, also write P to denote the subset of Σ ∗ that satisfies the promise;analogously, we write P n,f ,...,f h to denote those instances of size n that satisfy the promise. For h X i ∈ P , let P ( h X i ) ∈ { , } denote the desired output on input X . We also use the notation P = ( P , P ), where P j = {h X i ∈ P : P ( h X i ) = j } .We say that P = ( P n,f ,...,f h ) n ∈ N is many-one reducible to P ′ = ( P ′ m,f ′ ,...,f ′ h ′ ) m ∈ N if ∃ p , . . . , p h ′ ,where each p j is a real ( h + 1)-variate polynomial, such that ∀ n ∈ N , ∃ g n : P n,f ,...,f h → P ′ m,f ′ ,...,f ′ h ′ such that (1) m = p ( n, f ( n ) , . . . , f h ( n )), (2) f ′ j ( m ) = p j ( n, f ( n ) , . . . , f h ( n )), ∀ j , and (3) such that P ( h X i ) = P ′ ( g n ( h X i )), ∀h X i ∈ P n,f ,...,f h . If ( g n ) n ∈ N is computable in deterministic logspace (resp.uniform NC , uniform AC ), we write P ≤ m L P ′ (resp. P ≤ m NC P ′ , P ≤ m AC P ′ ). For a complexityclass C , we say that P ′ is C -complete if (1) P ′ ∈ C and (2) P ≤ m L P ′ , ∀ P ∈ C .We assume that the reader has familiarity with quantum computation and the theory of quan-tum information; see, for instance, [24, 33, 54] for an introduction. A quantum system, on s qubits,that is in a pure state is described by a unit vector | ψ i in the 2 s -dimensional Hilbert space C s . A mixed state of the same system is described by some ensemble { ( p i , | ψ i i ) : i ∈ I } , for some indexset I , where p i ∈ [0 ,
1] denotes the probability of the system being in the pure state | ψ i i ∈ C s ,and P i p i = 1. This ensemble corresponds to the density matrix A = P i p i | ψ i ih ψ i | ∈ Den(2 s ).Of course, many distinct ensembles correspond to the density matrix A ; however, all ensemblesthat correspond to a particular density matrix will behave the same, for our purposes (see, forinstance, [33, Section 2.4] for a detailed discussion of this phenomenon).Let T( n, m ) denote the set of all superoperators of the form Φ : Mat( n ) → Mat( m ) (i.e., Φ is a C -linear map from the C -vector space Mat( n ) to the C -vector space Mat( m )). Let T( n ) = T( n, n )and let n ∈ T( n ) denote the identity operator. Consider some Φ ∈ T( n, m ). We say that Φ is positive if, ∀ A ∈ Pos( n ), we have Φ( A ) ∈ Pos( m ). We say that Φ is completely-positive if Φ ⊗ r is positive, ∀ r ∈ N , where ⊗ denotes the tensor product. We say that Φ is trace-preserving iftr(Φ( A )) = tr( A ), ∀ A ∈ Mat( n ). If Φ is both completely-positive and trace-preserving, then we sayΦ is a quantum channel ; let Chan( n, m ) = { Φ ∈ T( n, m ) : Φ is a quantum channel } denote the setof all such channels, and let Chan( n ) = Chan( n, n ).Let vec denote the usual vectorization map that takes a matrix A ∈ Mat( n ) to the vectorvec( A ) ∈ C n consisting of the entries of A (in some fixed order). For Φ ∈ T( n ), let K (Φ) ∈ Mat( n )denote the natural representation of Φ, which is defined to be the (unique) matrix for whichvec(Φ( A )) = K (Φ)vec( A ), ∀ A ∈ Mat( n ). We now briefly recall the definitions of several needed space-bounded quantum complexity classes.We begin by noting that, in many of the previous papers that considered space-bounded quantumcomputation [23, 31, 35, 45, 49–52], the quantum Turing machine model was used to define thevarious complexity classes of interest. Arguably, this is the “natural” model to be used to definethese classes. However, as the (equivalent) model of uniformly generated quantum circuits are,arguably, more familiar to quantum complexity theorists and physicists, we state these definitionsusing quantum circuits. We emphasize that all of the results in this paper apply to both the uniformquantum circuit model and the quantum Turing machine model.
Definition 6.
A (unitary) quantum circuit is a sequence of quantum gates, each of which is amember of some fixed set of gates that is universal for quantum computation (e.g., { H,CNOT,T } ).We say that a family of quantum circuits { Q w : w ∈ P } is DSPACE ( s ( n ))-uniform if there is adeterministic Turing machine that, on any input w ∈ P , runs in space O ( s ( | w | )) (and hence time2 O ( s ( | w | )) ), and outputs a description of Q w . 5 efinition 7. Consider functions c, k : N → [0 ,
1] and s : N → N , with s ( n ) = Ω(log n ), all of whichare computable in DSPACE ( s ( n )). Let Q U SPACE ( s ( n )) c ( n ) ,k ( n ) denote the collection of all promiseproblems P = ( P , P ) such that there is a DSPACE ( s ( n ))-uniform family of (unitary) quantumcircuits { Q w : w ∈ P } , where Q w acts on h w = O ( s ( | w | )) qubits and has 2 O ( s ( | w | )) gates, which hasthe following properties. The circuit Q w is applied to h w qubits that were initialized in the all-zerosstate | h w i , after which the first qubit is measured in the standard basis. If the result is 1, then wesay Q w accepts w ; if, instead, the result is 0, then we say Q w rejects w . Let Π = | ih | ⊗ I hw − .We require that the following conditions hold: Completeness: w ∈ P ⇒ Pr[ Q w accepts w ] = k Π Q w | h w ik ≥ c ( | w | ). Soundness: w ∈ P ⇒ Pr[ Q w accepts w ] = k Π Q w | h w ik ≤ k ( | w | ).Note that, in the preceding definitions, Q w has 2 O ( s ( | w | )) gates (this requirement is immediatelyforced by the uniformity condition). That is to say, in our definition of quantum space s ( n ), werequire that the computation also runs in time 2 O ( s ( | w | )) . We refer the reader to the excellent surveypaper by Saks [39] for a thorough discussion of the desirability of requiring that space-bounded probabilistic computations run in time at most exponential in their space bound, as well as to, forinstance, [31, 49, 51] for discussions of the analogous issue for quantum computations. Definition 8.
We then define unitary quantum space s ( n ) for a variety of types of error, as follows. Two-sided bounded-error: BQ U SPACE ( s ( n )) = Q U SPACE ( s ( n )) , . One-sided bounded-error: RQ U SPACE ( s ( n )) = Q U SPACE ( s ( n )) , . One-sided unbounded-error: NQ U SPACE ( s ( n )) = S c : N → (0 , Q U SPACE ( s ( n )) c ( n ) , .Then, for X ∈ { B , R , N } , let XQ U L = XQ U SPACE (log n ) denote unitary quantum logspace witherror-bounds specified by the modifier X .Note that the particular choice of universal gateset does not affect the definition of the two-sided bounded-error classes BQ U SPACE ( s ( n )) (in particular, of BQ U L ) due to the space-efficientversion [31] of the Solovay-Kitaev theorem; however, the definitions of the one-sided (bounded andunbounded) error classes are, potentially, gateset dependent. Furthermore, note that the class BQ U L would remain the same if defined as BQ U L = Q U SPACE (log n ) c ( n ) ,k ( n ) for any c, k for which ∃ q : N → N > , where q ( n ) = poly ( n ), such that c ( n ) − k ( n ) ≥ q ( n ) , ∀ n [17]; an analogous claimholds for RQ U L [49].We next consider general space-bounded quantum computation. Most basically, we wish todefine a model of quantum computation that allows intermediate quantum measurements . That isto say, rather than considering a purely unitary quantum computation in which a single measure-ment is performed at the end of the computation, we now allow measurements to be performedthroughout the computation, and for the the results of those measurements to be used to controlthe computation. As we wish for our main result (the fact that unitary and general space-boundedquantum computation are equivalent in power) to be as strong as possible, we wish to define amodel of general space-bounded quantum computation that is as powerful as possible. To thatend, we will define a space-bounded variant of the general quantum circuit model considered inthe classic paper of Aharonov, Kitaev, and Nisan [2], in which gates are now arbitrary quantumchannels . 6 efinition 9. A general quantum circuit on h qubits is a sequence Φ = (Φ , . . . , Φ t ) of quantumchannels, where each Φ j ∈ Chan(2 h ). By slight abuse of notation, we use Φ to denote the elementΦ t ◦ · · · ◦ Φ ∈ Chan(2 h ) obtained by composing the individual gates of the circuit in order. Wesay that a family of general quantum circuits { Φ w = (Φ w, , . . . , Φ w,t w ) : w ∈ P } is DSPACE ( s ( n ))-uniform if there is a deterministic Turing machine that, on any input w ∈ P , runs in space O ( s ( | w | ))(and hence time 2 O ( s ( | w | )) ), and outputs a description of Φ w ; to be precise, a description of Φ w consists of the entries of each K (Φ w,j ), where we require that K (Φ w,j ) ∈ d Mat(2 h ).Note that the operation of applying a unitary transformation is a special case of a quantumchannel, and so the general quantum circuit model extends the ordinary (unitary) quantum circuitmodel. Moreover, note that the process of performing any (partial or full) quantum measurementin the computational basis is described by a quantum channel and that the form of the precedingdefinition allows the results of intermediate measurements to be used to control which operationsare applied at later stages of the computation (this can be accomplished by using a subset of thequbits as classical bits to store the results of earlier measurements, thereby making these resultsavailable to gates that appear later in the computation). It is necessary to establish some reasonablerestriction on the complexity of computing a description of each gate of the circuit, as we do notwish to unreasonably increase the power of the model by allowing, for example, non-computablenumbers to be used in defining each gate (see, for instance, [31, 36, 37, 51] for further discussion ofthis issue). Definition 10.
Consider functions c, k : N → [0 ,
1] and s : N → N , with s ( n ) = Ω(log n ), allof which are computable in DSPACE ( s ( n )). Let QSPACE ( s ( n )) c ( n ) ,k ( n ) denote the collection ofall promise problems P = ( P , P ) such that there is a DSPACE ( s ( n ))-uniform family of generalquantum circuits { Φ w : w ∈ P } , where Φ w acts on h w = O ( s ( | w | )) qubits and has 2 O ( s ( | w | )) gates,which has the following properties. The circuit Φ w is applied to h w qubits that were initialized inthe all-zeros state | h w i , after which the first qubit is measured in the standard basis. If the resultis 1, then Φ w accepts w ; otherwise, Φ w rejects w . Let ⊗ tr ∈ Chan(2 h w ,
2) denote the operationin which all qubits except the first qubit are traced out . Completeness: w ∈ P ⇒ Pr[Φ w accepts w ] = |h | ( ⊗ tr)(Φ w ( | h w ih h w | )) | i| ≥ c ( | w | ). Soundness: w ∈ P ⇒ Pr[Φ w accepts w ] = |h | ( ⊗ tr)(Φ w ( | h w ih h w | )) | i| ≤ k ( | w | ).We note that this general quantum circuit model is equivalent to the space-bounded (general)quantum Turing machine model of Watrous [51] (modulo a small issue concerning the preciseclass of allowed transition amplitudes, in which our definition is slightly more restrictive). Wealso note that the results of this paper would apply to any “reasonable” variant of space-boundedquantum computation that is classically controlled, which includes all of the “standard” variantsthat have been considered (see, for instance, [31, 35, 45, 51]). We refer the reader to [31, Section 2]for a thorough discussion of the various models of space-bounded quantum computation, and, inparticular, of the reasonableness of requiring classical control; we also discuss the issue of classicalvs. quantum control later in Section 6. Definition 11.
We then general quantum space s ( n ) for each error-type, analogously to the unitary case of Definition 8. BQSPACE ( s ( n )) = QSPACE ( s ( n )) , , BQL = BQSPACE (log n ), RQSPACE ( s ( n )) = QSPACE ( s ( n )) , , etc.Note that the particular choice of constants and that appear in the definition of BQSPACE are arbitrary, as the completeness and soundness parameters can straightforwardly be amplified;7 similar statement holds for
RQSPACE . Finally, we define a space-bounded variant of quantumMerlin-Arthur proof systems, essentially following [18].
Definition 12.
Consider functions c, k : N → [0 ,
1] and s : N → N , with s ( n ) = Ω(log n ), all ofwhich are computable in DSPACE ( s ( n )). Let s ( n ) − bounded − QMA c ( n ) ,k ( n ) denote the collection ofall promise problems P = ( P , P ) such that there is a DSPACE ( s ( n ))-uniform family of (unitary)quantum circuits { V w : w ∈ P } , where V w acts on m w + h w = O ( s ( | w | )) qubits and has 2 O ( s ( | w | )) gates, which has the following properties. Let Π = | ih | ⊗ I mw + hw − and let Ψ m w denote the setof m w -qubit states. For each w ∈ P , the verification circuit V w is applied to the state | ψ i ⊗ | h w i ,where | ψ i ∈ Ψ m w is a (purported) proof of the fact that w ∈ P . Then, the first qubit is measuredin the standard basis. If the result is 1, then w is accepted ; otherwise, w is rejected . Completeness: w ∈ P ⇒ ∃| ψ i ∈ Ψ m w , Pr[ V w accepts w, | ψ i ] = k Π V w ( | ψ i ⊗ | h w i ) k ≥ c ( | w | ). Soundness: w ∈ P ⇒ ∀| ψ i ∈ Ψ m w , Pr[ V w accepts w, | ψ i ] = k Π V w ( | ψ i ⊗ | h w i ) k ≤ k ( | w | ). Definition 13.
We then define space-bounded
QMA , for a variety of types of error, as follows.
Bounded-error:Two-sided :
QMASPACE ( s ( n )) = s ( n ) − bounded − QMA , . Perfect completeness:
QMASPACE ( s ( n )) = s ( n ) − bounded − QMA , . Perfect soundness:
RQMASPACE ( s ( n )) = s ( n ) − bounded − QMA , . Unbounded-error:Perfect completeness:
PreciseQMASPACE ( s ( n )) = S k : N → [0 , s ( n ) − bounded − QMA ,k ( n ) . Perfect soundness:
NQMASPACE ( s ( n )) = S c : N → (0 , s ( n ) − bounded − QMA c ( n ) , .Let QMAL = QMASPACE (log n ), QMAL = QMASPACE (log n ), etc. We define the following well-conditioned versions of the standard
DET ∗ -complete problems [12]. Definition 14.
Consider functions m : N → N , κ : N → R ≥ , and ǫ : N → R > . For a sequence ofmatrices A , . . . , A m , and for indices j , j , where 1 ≤ j ≤ j ≤ m , let A j ,j = j Q j = j A j .(i) DET n,κ,ǫ − Input : A ∈ d Mat( n ), b ∈ R ≤ . Promise : σ ( A ) ≤ σ n ( A ) ≥ κ ( n ) , | det( A ) | ∈ [ κ ( n ) − n , e b − ǫ ( n ) ] ∪ [ e b , Output : 1 if | det( A ) | ≥ e b , 0 otherwise.(ii) DET + n,κ,ǫ − Input : H ∈ d Pos( n ), b ∈ R ≤ . Promise : σ ( H ) ≤ σ n ( H ) ≥ κ ( n ) , det( H ) ∈ [ κ ( n ) − n , e b − ǫ ( n ) ] ∪ [ e b , Output : 1 if det( H ) ≥ e b , 0 otherwise. 8iii) MATPOW n,m,κ,ǫ − Input : A ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ R ≥ . Promise : σ ( A j ) ≤ κ ( n ) , ∀ j ∈ { , . . . , m } , | A m [ s, t ] | ∈ [0 , b − ǫ ( n )] ∪ [ b, κ ( n )]. Output : 1 if | A m [ s, t ] | ≥ b , 0 otherwise.(iv) ITMATPROD n,m,κ,ǫ − Input : A , . . . , A m ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ R ≥ . Promise : σ ( A j ,j ) ≤ κ ( n ) for 1 ≤ j ≤ j ≤ m , | A ,m [ s, t ] | ∈ [0 , b − ǫ ( n )] ∪ [ b, κ ( n )]. Output : 1 if | A ,m [ s, t ] | ≥ b , 0 otherwise.(v) ITMATPROD ≥ n,m,κ,ǫ − Input : A , . . . , A m ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ R ≥ . Promise : σ ( A j ,j ) ≤ κ ( n ) for 1 ≤ j ≤ j ≤ m , A ,m [ s, t ] ∈ [0 , b − ǫ ( n )] ∪ [ b, κ ( n )]. Output : 1 if A ,m [ s, t ] ≥ b , 0 otherwise.(vi) SUMITMATPROD n,m,κ,ǫ − Input : A , . . . , A m ∈ d Mat( n ), E ⊆ { , . . . , n } , b ∈ R ≥ . Promise : σ ( A j ,j ) ≤ κ ( n ) for 1 ≤ j ≤ j ≤ m , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ( s,t ) ∈ E A ,m [ s, t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∈ [0 , b − ǫ ( n )] ∪ [ b, | E | κ ( n )]. Output : 1 if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ( s,t ) ∈ E A ,m [ s, t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ b , 0 otherwise.(vii) MATINV n,κ,ǫ − Input : A ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ R ≥ . Promise : σ ( A ) ≤ σ n ( A ) ≥ κ ( n ) , | A − [ s, t ] | ∈ [0 , b − ǫ ( n )] ∪ [ b, κ ( n )]. Output : 1 if | A − [ s, t ] | ≥ b , 0 otherwise.(viii) MATINV + n,κ,ǫ − Input : H ∈ d Pos( n ), s, t ∈ { , . . . , n } , b ∈ R ≥ . Promise : σ ( H ) ≤ σ n ( H ) ≥ κ ( n ) , | H − [ s, t ] | ∈ [0 , b − ǫ ( n )] ∪ [ b, κ ( n )]. Output : 1 if | H − [ s, t ] | ≥ b , 0 otherwise.Note that, with the exception of the problem DET (and
DET + ), each of the above problemsare defined such that they correspond to approximating some quantity with additive error ǫ/ MATINV involves determining if | A − [ s, t ] | ≤ b − ǫ or | A − [ s, t ] | ≥ b . To clarify ourdefinition of DET , observe that this problem, which involves determining if | det( A ) | ≤ e b − ǫ or | det( A ) | ≥ e b , is equivalent to the problem of determining if ln( | det( A ) | ) ≤ b − ǫ or ln( | det( A ) | ) ≥ b .In other words, we have defined DET such that it corresponds to obtaining an approximation ofln( | det( A ) | ) with additive error ǫ/
2; this is equivalent to obtaining a e ± ǫ multiplicative approxi-mation of | det( A ) | . As we will see in Section 3.1 and Section 3.3, this is the “correct” definition of DET , in the sense that it is the version of the determinant problem that most closely correspondsto the other linear-algebraic problems (matrix powering, matrix inversion, etc.) defined above.Moreover, note that the problems as stated above are somewhat “over parameterized.” Forexample, if h A, s, t, b i ∈
MATINV n,κ ( n ) ,ǫ − ( n ) , then h ǫ ( n ) A, s, t, ǫ − ( n ) b i ∈ MATINV n,κ ( n ) ǫ − ( n ) , and MATINV ( h A, s, t, b i ) = MATINV ( h ǫ ( n ) A, s, t, ǫ − ( n ) b i ). These additional parameters are convenientas they allow us to express certain results more cleanly.9 efinition 15. For each promise problem P n,m,κ,ǫ − in Definition 14, we define poly -conditioned- P to be the promise problem P n,n O (1) ,n O (1) ,n O (1) . For example, poly -conditioned- DET + Input : H ∈ d Pos( n ), b ∈ R ≤ . Promise : σ ( H ) ≤ σ n ( H ) ≥ n − O (1) , det( H ) ∈ [ n − O ( n ) , e b − n − O (1) ] ∪ [ e b , Output : 1 if det( H ) ≥ e b , 0 otherwise. U L Completeness
We now show that all of the above poly -conditioned problems are BQ U L -complete. By [18, The-orem 13], poly -conditioned- MATINV + is BQ U L -complete (note that the notation used in this pa-per to express the promised condition number differs from the notation used in [18]). There-fore, it suffices to show that, for each such poly -conditioned- P , we have poly -conditioned- P ≤ m L poly -conditioned- MATINV + and poly -conditioned- MATINV + ≤ m L poly -conditioned- P . Remark.
We note that reductions between the standard versions of these problems (i.e., wherethere is no assumption of being well-conditioned) are well-known [5, 12, 13, 29, 46–48]. However,these reductions, generally, do not preserve the property of being poly -conditioned; therefore, wemust exhibit reductions that are rather different from the “standard” reductions.We will prove several lemmas that exhibit reductions between particular problems in Defini-tion 14. The proofs of these lemmas share a common structure: for a pair of promise problems P , P ′ , we show how to transform an instance w ∈ P to an instance f ( w ) ∈ P ′ such that the re-duction function f preserves the answer (i.e., P ( w ) = P ′ ( f ( w ))) and also preserves the property ofbeing well-conditioned. Note that P ≤ m L P ′ ⇒ poly -conditioned- P ≤ m L poly -conditioned- P ′ . In thefollowing, we assume that m ( n ), κ ( n ), and ǫ ( n ) − can be computed to O (log n ) bits of precision inuniform AC .Note that, the proof (which appears in Section 3.2) of Theorem 1 (which states the equivalencebetween unitary and general quantum space) only requires Lemmas 21 to 23. The remaininglemmas are used to prove the other main results of this paper. Lemma 16.
DET ≤ m AC SUMITMATPROD .Proof.
Consider some h H, b i ∈
DET + n,κ,ǫ − . By the promise, H ∈ d Pos( n ), λ ( H ) = σ ( H ) ≤
1, and λ n ( H ) = σ n ( H ) ≥ κ ( n ) − , which implies σ ( I − H ) = λ ( I − H ) = 1 − λ n ( H ) ≤ − κ ( n ) − < H ) = − ∞ P k =1 ( I − H ) k k , where here log( H ) denotes the matrix logarithm. Recall that,as a consequence of Jacobi’s formula, ln(det( H )) = tr(log( H )).For m ∈ N ≥ , let S m = m P k =1 ( I − H ) k k , let R m = ∞ P k = m +1 ( I − H ) k k = − log( H ) − S m , and let D m ∈ d Mat( m ) denote the diagonal matrix where D m [ k, k ] = k . Let b l = ⌊ ⌊ κ ( n ) ⌋ ) ⌋ , let b A = I n b l ⊕ ( − D m ⊗ ( I − H )) ∈ d Mat( n b l + nm ), and, for k ∈ { , . . . , m } , let b A k = I n ( b l + k − ⊕ ( I m +1 − k ⊗ ( I − H )) ∈ d Mat( n b l + nm ). Then b A ,m = m Y j =1 b A j = I n b l ⊕ m M k =1 − ( I − H ) k k ! . Let E m = { ( d, d ) : d ∈ { , . . . , n b l + nm }} . We then have X ( s,t ) ∈ E m b A ,m [ s, t ] = tr( b A ,m ) = tr( I n b l ) − m X k =1 tr (cid:18) ( I − H ) k k (cid:19) = n b l − tr( S m ) = n b l +ln(det( H ))+tr( R m ) . ≤ j ≤ j ≤ m , we have σ ( b A j ,j ) ≤ max (cid:18) σ ( I n b l ) , max k ∈{ ,...,j − j } σ (( I − H ) k ) (cid:19) = 1 . As shown above, σ ( I − H ) ≤ − κ ( n ) − , which implies σ ( R m ) = σ ∞ X k = m +1 ( I − H ) k k ! ≤ ∞ X k = m +1 ( σ ( I − H )) k k ≤ ∞ X k = m +1 (1 − κ ( n ) − ) k k ≤ κ ( n ) (cid:18) − κ ( n ) (cid:19) m +1 . If m ≥ κ ( n ) ln(2 nκ ( n ) ǫ ( n ) − ), thentr( R m ) ≤ nσ ( R m ) ≤ nκ ( n ) (cid:18) − κ ( n ) (cid:19) κ ( n ) ln(2 nκ ( n ) ǫ ( n ) − ) ≤ nκ ( n ) (cid:18) e (cid:19) ln(2 nκ ( n ) ǫ ( n ) − ) = 12 ǫ ( n ) . Let b m = ⌈ κ ( n ) ⌉⌊ ⌊ nκ ( n ) ǫ ( n ) − ⌋ ) ⌋ ≥ κ ( n ) ln(2 nκ ( n ) ǫ ( n ) − ) and b E = E b m . Note thattr( R m ) ≥
0. We then have, n b l + ln(det( H )) ≤ X ( s,t ) ∈ b E b A ,m [ s, t ] = n b l + ln(det( H )) + tr( R b m ) ≤ n b l + ln(det( H )) + 12 ǫ ( n ) . If det( H ) ≥ e b , then P ( s,t ) ∈ b E b A ,m [ s, t ] ≥ n b l + b ; if det( H ) ≤ e b − ǫ ( n ) , then P ( s,t ) ∈ b E b A ,m [ s, t ] ≤ n b l + b − ǫ ( n ). Let b b = n b l + b . Therefore, h b A , . . . , b A b m , b E, b b i ∈ SUMITMATPROD n ( b l + b m ) , b m, , ǫ − ( n ) and DET ( h H, b i ) = SUMITMATPROD ( h b A , . . . , b A b m , b E, b b i ). Lemma 17.
SUMITMATPROD ≤ m AC ITMATPROD .Proof.
Consider some h A , . . . , A m , E, b i ∈ SUMITMATPROD n,m,κ,ǫ − . Let T c,d ∈ d Mat( n ) denotethe permutation matrix corresponding to interchanging elements c, d ∈ { , . . . , n } and leaving allother elements fixed. For j ∈ { , . . . , m } , let b A j = L ( s,t ) ∈ E T ,t A j T ,s ∈ d Mat( n | E | ) . Let R ∈ d Mat( | E | )be defined such that R r,c = 1 if r = c or r = 1, and R r,c = 0 otherwise; let b A = R ⊗ I n and b A m +1 = b A † . We then have b A ,m +1 [1 ,
1] = P ( s,t ) ∈ E A ,m [ s, t ]. Notice that σ ( b A ) = σ ( b A m +1 ) = σ ( R ) σ ( I n ) ≤ p | E | , which implies σ (cid:16) b A j ,j (cid:17) ≤ | E | σ (cid:0) A max( j , , min( j ,m ) (cid:1) ≤ | E | κ ( n ) ≤ n κ ( n ) , for 0 ≤ j ≤ j ≤ m + 1 . Let b s = b t = 1 and b b = b . Then h b A , . . . , b A m +1 , b s, b t, b b i ∈ ITMATPROD n ,m +2 , n κ ( n ) ,ǫ − ( n ) and SUMITMATPROD ( h A , . . . , A m , E, b i ) = ITMATPROD ( h b A , . . . , b A m +1 , b s, b t, b b i ). Lemma 18.
ITMATPROD ≤ m AC ITMATPROD ≥ .Proof. Consider some h A , . . . , A m , s, t, b i ∈ ITMATPROD n,m,κ,ǫ − . Let χ t ∈ R n denote the vectorwhere χ t [ t ] = 1 and χ t [ k ] = 0 ∀ k = t . For j ∈ { , . . . , m + 1 } , we define b A j = A j , j ≤ mχ t χ † t , j = m + 1 A † m +2 − j , j ≥ m + 211e then have b A , m +1 [ s, s ] = A ,m [ s, t ] A ,m [ s, t ] = | A ,m [ s, t ] | . Consider j , j such that 1 ≤ j ≤ j ≤ m + 1. If j ≤ m + 1 ≤ j , then σ ( b A j ,j ) = σ ( A j ,m χ t χ † t A † j ,m ) ≤ σ ( A j ,m ) σ ( χ t χ † t ) σ ( A † j ,m ) ≤ κ ( n ) . If j < m + 1, then σ ( b A j ,j ) = σ ( A j ,j ) ≤ κ ( n ); finally, if j > m + 1, then σ ( b A j ,j ) = σ ( A j − m − ,j − m − ) ≤ κ ( n ).Let b s = b t = s and b b = b . Then h b A , . . . , b A m +1 , b s, b t, b b i ∈ ITMATPROD ≥ n, m +1 ,κ ,ǫ − and ITMATPROD ( h A , . . . , A m , s, t, b i ) = ITMATPROD ( h b A , . . . , b A m +1 , b s, b t, b b i ). Lemma 19.
ITMATPROD ≥ ≤ m AC DET .Proof.
Consider some h A , . . . , A m , s, t, b i ∈ ITMATPROD ≥ n,m,κ,ǫ − . Let Y ∈ d Mat( nm + n ) consist of n × n blocks, where the blocks immediately above the main diagonal blocks are given by A , . . . , A m ,and all other entries are 0. Let B = I nm + n − Y ∈ d Mat( nm + n ) and consider B − . We partition B − into n × n blocks, where for r, c ∈ { , . . . , m + 1 } , C r,c ∈ d Mat( n ) denotes the block in block-row r and block-column c . It is straightforward to verify that C r,c = A r,c − , r > cI n , r = c n , r < c For k ∈ { , . . . , nm + n } , let χ k ∈ R nm + n denote the vector that is 1 in entry k and 0 elsewhere;let v = χ s and let u = χ nm + t . Notice that det( B ) = 1. By the matrix determinant lemma,det( B + uv † ) = (1 + v † B − u ) det( B ) = 1 + B − [ s, nm + t ] = 1 + C ,m +1 [ s, t ] = 1 + A ,m [ s, t ] . Next, observe that σ ( B + uv † ) ≤ σ ( uv † ) + σ ( I ) + σ ( Y ) ≤ j σ ( A j ) ≤ κ ( n ) . Notice that B − = m P j =0 Y j , which implies σ ( B − ) ≤ m X j =0 σ ( Y j ) ≤ m X j =1 (cid:18) max k ∈{ ,...,m − j +1 } σ ( A k,k + j − ) (cid:19) ≤ m X j =1 κ ( n ) = 1 + mκ ( n ) . By the Sherman-Morrison formula, ( B + uv † ) − = B − ( I − (1 + v † B − u ) − uv † B − ). Recall that,by the promise, v † B − u = A ,m [ s, t ] ∈ R ≥ . Therefore, σ (( B + uv † ) − ) ≤ σ ( B − )( σ ( I ) + σ ((1 + v † B − u ) − uv † ) σ ( B − )) ≤ (1 + mκ ( n ))(2 + mκ ( n )) . This implies σ nm + n ( B ) = σ ( B − ) − ≥ ((1 + mκ ( n ))(2 + mκ ( n ))) − . Let b l = ⌊ ⌊ κ ( n ) ⌋ ) ⌋ and let b B = e − b l ( B + uv † ) ∈ d Mat( nm + n ). Then, for j ∈ { , . . . , nm + n } , σ j ( b B ) = e − b l σ j ( B ); inparticular, σ ( b B ) ≤ σ nm + n ( b B ) ≥ (2 + mκ ( n )) − . Moreover, | det( b B ) | = | e − b l ( nm + n ) det( B + uv † ) | = | e − b l ( nm + n ) (1 + A ,m [ s, t ]) | = e − b l ( nm + n ) (1 + A ,m [ s, t ]) . b a = ln(1+ b − ǫ ( n )) − b l ( nm + n ) and b b = ln(1+ b ) − b l ( nm + n ). If A ,m [ s, t ] ≥ b , then | det( b B ) | ≥ e b b ;if A ,m [ s, t ] ≤ b − ǫ ( n ), then | det( b B ) | ≤ e b a . We have b b − b a = ln (cid:18) b b − ǫ ( n ) (cid:19) = ln (cid:18) ǫ ( n )1 + b − ǫ ( n ) (cid:19) ≥ ln (cid:18) ǫ ( n )1 + κ ( n ) (cid:19) ≥ ǫ ( n )2(1 + κ ( n )) . Therefore, h b B, b b i ∈ DET nm + m, (2+ mκ ( n )) ,ǫ − ( n )(2+2 κ ( n )) and ITMATPROD ( h A , . . . , A m , s, t, b i ) = DET ( h b B, b b i ). Lemma 20.
DET ≤ m NC DET + .Proof. Consider h A, b i ∈
DET n,κ,ǫ − . Let b H = AA † ∈ d Pos( n ) and b b = 2 b . Then, det( b H ) = | det( A ) | and σ j ( b H ) = σ j ( A ). Therefore, h b H, b b i ∈ DET n,κ , ǫ − and DET ( h A, b i ) = DET ( h b H , b b i ). Lemma 21.
ITMATPROD ≤ m AC MATPOW .Proof.
Following [12], consider h A , . . . , A m , s, t, b i ∈ ITMATPROD n,m,κ,ǫ − . Let b A ∈ d Mat( nm + n )consist of n × n blocks, where the blocks immediately above the main diagonal blocks are givenby A , . . . , A m . Let b s = s , b t = nm + t , b b = b . Then b A m [ b s, b t ] = A ,m [ s, t ]. Therefore, h b A, b s, b t, b b i ∈ MATPOW nm + n,m,κ ( n ) ,ǫ − ( n ) and ITMATPROD ( h A , . . . , A m , s, t, b i ) = MATPOW ( h b A, b s, b t, b b i ). Lemma 22.
MATPOW ≤ m AC MATINV .Proof.
Following [12], consider h A, s, t, b i ∈
MATPOW n,m,κ,ǫ − . Let Y ∈ d Mat( nm + n ) consist of n × n blocks, where the blocks immediately above the main diagonal blocks are all given by A . Let Z = I nm + n − Y and observe that Z − consists of n × n blocks, where the submatrix in block-row r and block-column c is given by A c − r when c ≥ r , and 0 n when c < r . Let b s = s and b t = nm + t ;then Z − [ b s, b t ] = A m [ s, t ]. Moreover, σ ( Z ) ≤ σ ( I nm + n ) + σ ( Y ) ≤ κ ( n ). Furthermore, σ ( Z − ) ≤ m X j =1 σ ( A j ) ≤ m X j =1 κ ( n ) ≤ mκ ( n ) . This implies σ nm + n ( Z ) = σ ( Z − ) − ≥ (1 + mκ ( n )) − . Let b Z = ⌈ κ ( n ) ⌉ Z and b b = ⌈ κ ( n ) ⌉ b . We then conclude that h b Z, b s, b t, b b i ∈ MATINV nm + n, (1+ mκ ( n )) ⌈ κ ( n ) ⌉ , ⌈ κ ( n ) ⌉ − ǫ − ( n ) and MATPOW ( h A, s, t, b i ) = MATINV ( h b Z, b s, b t, b b i ). Lemma 23.
MATINV ≤ m NC MATINV + .Proof. Consider h A, s, t, b i ∈
MATINV κ,ǫ − . Let b H = (cid:18) A † A − A † − A I (cid:19) ∈ d Pos(2 n ). Then b H − =3 (cid:18) A † A ) − A − ( A † ) − I (cid:19) . Moreover, σ ( b H ) ≤ σ n ( b H ) ≥ ( σ n ( A )) ≥ (3 κ ( n )) − . Let b s = s , b t = t + n , and b b = 3 b . Then b H − [ b s, b t ] = 3 A − [ s, t ]. Therefore, h b H, b s, b t, b b i ∈ MATINV +2 n, (3 κ ( n )) , (3 ǫ ( n )) − and MATINV ( h A, s, t, b i ) = MATINV ( h H, b s, b t, b b i ). Lemma 24.
MATINV + ≤ m AC SUMITMATPROD . roof. Consider h H, s, t, b i ∈
MATINV + κ,ǫ − . For m ∈ N , we have m X j =0 ( I − H ) j = H − ( I − ( I − H ) m +1 ) . Let b m = ⌈ κ ( n ) ⌉⌊ ⌊ κ ( n ) ǫ ( n ) − ⌋ ) ⌋ . For j ∈ { , . . . , b m } , let b A j = I jn ⊕ ( I b m − j +1 ⊗ ( I − H )) ∈ d Mat( n b m + n ). For 1 ≤ j ≤ j ≤ b m , we have σ ( b A j ,j ) = max j ∈{ ,...,j − j +1 } σ (( I − H ) j ) = 1. Let b E = { ( s + jn, t + jn ) : j ∈ { , . . . , b m }} . We then have X ( b s, b t ) ∈ b E b A , b m [ b s, b t ] = b m X j =0 ( I − H ) j [ s, t ] = ( H − ( I − ( I − H ) b m +1 ))[ s, t ] . This implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( b s, b t ) ∈ b E b A , b m [ b s, b t ] (cid:12)(cid:12)(cid:12)(cid:12) − | H − [ s, t ] | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ( H − ( I − H ) b m +1 )[ s, t ] | ≤ σ ( H − ( I − H ) b m +1 ) ≤ σ ( H − )( σ ( I − H )) b m +1 ≤ κ ( n ) (cid:18) − κ ( n ) (cid:19) b m +1 ≤ ǫ ( n ) . Let b b = b − ǫ ( n ). We then conclude that h b A , . . . , b A b m , b E, b b i ∈ SUMITMATPROD n b m + n, b m, , ǫ − ( n ) and MATINV ( h H, s, t, b i ) = SUMITMATPROD ( h b A , . . . , b A b m , b E, b b i ).We now prove Theorem 4 from Section 1.1, which we restate here for convenience. Theorem 4.
Each poly -conditioned promise problem given in Definition 15 is BQ U L -complete.Proof. By [18, Theorem 13], poly -conditioned-
MATINV + is BQ U L -complete. By the preceding lem-mas, we have MATINV + ≤ m AC SUMITMATPROD ≤ m AC ITMATPROD ≤ m AC MATPOW ≤ m AC MATINV ≤ m NC MATINV + and DET + ≤ m AC SUMITMATPROD ≤ m AC ITMATPROD ≤ m AC ITMATPROD ≥ ≤ m AC DET ≤ m NC DET + . Note that ≤ m AC or ≤ m NC reducibility implies ≤ m L reducibility; further, note that P ≤ m L P ′ ⇒ poly -conditioned- P ≤ m L poly -conditioned- P ′ . Therefore, we conclude that each such poly -conditionedproblem is BQ U L -complete. U SPACE vs. BPSPACE vs. QMASPACE
In this section, we explore the relationships between the classes
BQSPACE ( s ( n )) , BQ U SPACE ( s ( n )) , BPSPACE ( s ( n )) , and QMASPACE ( s ( n )). First, we consider the case in which s ( n ) = log n . While,trivially, BPL ⊆ BQL , it is not obvious, a priori , that
BPL ⊆ BQ U L . To the best of our knowledge,the strongest partial result in this direction is the classic result of Watrous [49, Theorem 4.12], whichshowed that BPL is contained in a variant of BQ U L in which there is no bound on the running timeof the QTM. 14y Theorem 4, poly -conditioned- MATPOW ∈ BQ U L . As we next observe, this implies BPL ⊆ BQ U L and, more strongly, BQL = BQ U L . Of course, the statement BQL = BQ U L immediatelyimplies BPL ⊆ BQ U L ; nevertheless, we will first show, directly, that BPL ⊆ BQ U L . We then extendthese results to any (space-constructible) space bound s ( n ) = Ω(log n ), by use of a standard paddingargument. Proposition 25.
BPL ⊆ BQ U L .Proof. Suppose P = ( P , P ) ∈ BPL . Then there is some probabilistic TM M that recognizes P with two-sided error ≤ within time t ( n ) = n O (1) and space s ( n ) = O (log n ), for any input w ∈ P of length n = | w | . Let | M | denote the size of the finite control of M , let Γ denote thework-tape alphabet of M , and let c ( n ) = | M | ( n + 2)( s ( n )) | Γ | s ( n ) = n O (1) denote the number ofpossible configurations of M on inputs of length n . It is well-known that, for input w ∈ P , onemay construct, in deterministic space O (log( | w | )) a stochastic matrix A w ∈ d Mat( c ( n )) and values x w , y w ∈ { , . . . , c ( n ) } such that A tw [ x w , y w ] is precisely the probability that M accepts w within t steps [15,34]. Note that, as A w is stochastic, so is A tw , ∀ t ∈ N ; this implies σ ( A tw ) ≤ p c ( n ) = n O (1) .Therefore, h A w , x w , y w , i ∈ MATPOW c ( n ) ,t ( n ) , √ c ( n ) , and MATPOW ( h A w , x w , y w , i ) = P ( w ). ByTheorem 4, MATPOW c ( n ) ,t ( n ) , √ c ( n ) , ∈ BQ U L , which implies P ∈ BQ U L .By applying an analogous argument to general quantum Turing machines (where the stochasticmatrix that describes a single step of the computation of a probabilistic TM is replaced by thequantum channel that describes a single step of the computation of a quantum TM), we may thenshow that BQL ⊆ BQ U L (and, therefore, that BQL = BQ U L ); for completeness, we include sucha proof in Appendix A. Instead, in this section, we exhibit an analogous proof using the generalquantum circuits of Definition 9. Lemma 26. poly -conditioned-
ITMATPROD is BQL -hard.Proof.
Suppose P = ( P , P ) ∈ BQL . By definition, there is some L -uniform family of generalquantum circuits { Φ w = (Φ w, , . . . , Φ w,t w ) : w ∈ P } , where Φ w acts on h w = O (log | w | ) qubitsand has t w = | w | O (1) gates, such that if w ∈ P , then Pr[Φ w accepts w ] ≥ , and if w ∈ P , thenPr[Φ w accepts w ] ≤ . Without loss of generality we may, for convenience, assume that Φ w “cleans-up” its workspace at the end of the computation, by measuring the first qubit in the computationalbasis, and then forcing every other qubit to the state | i (by measuring each such qubit in thecomputational basis and, if the result 1 is obtained, flipping its value).Let d w = 2 h w = | w | O (1) . For j ∈ { , . . . , t w } , let A ( w ) j = K (Φ w,t w − j +1 ), and note that, byDefinition 9, A ( w ) j ∈ d Mat( d w ) and A ( w ) j can be constructed in deterministic space O (log( | w | )).Moreover, as Φ w,j ∈ Chan(2 h w ), for 1 ≤ j ≤ j ≤ t w , we have Φ w,t w − j +1 ◦ · · · ◦ Φ w,t w − j +1 ∈ Chan(2 h w ), which by [38, Theorem 1] implies the following bound on the largest singular value ofany partial product of the A ( w ) j σ ( A ( w ) j ,j ) = σ j Y j = j A ( w ) j = σ ( K (Φ w,t w − j +1 ◦ · · · ◦ Φ w,t w − j +1 )) ≤ p d w = n O (1) . Let x w = | h w − ih h w − | and y w = | h w ih h w | . By Definition 10,Pr[Φ w accepts w ] = t w Y j =1 A ( w ) j [ x w , y w ] = A ( w ) ,t w [ x w , y w ] .
15e then conclude that h A ( w ) , . . . , A ( w ) t w , x w , y w , i ∈ ITMATPROD n O (1) ,n O (1) ,n O (1) ,n O (1) and that ITMATPROD ( h A ( w ) , . . . , A ( w ) t w , x w , y w , i ) = P ( w ). Theorem 27.
BQL = BQ U L = QMAL .Proof.
Clearly, BQ U L ⊆ BQL . By Lemma 26, poly -conditioned-
ITMATPROD is BQL -hard; byTheorem 4, poly -conditioned-
ITMATPROD ∈ BQ U L , which implies BQL ⊆ BQ U L . By [18, Theorem18], QMAL = BQ U L .We now prove Theorem 1, our main result stated in the introduction, which we restate here forconvenience. Theorem 1.
For any space-constructible function s : N → N , where s ( n ) = Ω(log n ) , we have BQ U SPACE ( s ( n )) = BQSPACE ( s ( n )) = QMASPACE ( s ( n )) . Proof.
Clearly, BQ U SPACE ( s ( n )) ⊆ BQSPACE ( s ( n )). By [18, Theorem 18 and Theorem 26], QMASPACE ( s ( n )) = BQ U SPACE ( s ( n )). All that remains is to show that BQSPACE ( s ( n )) ⊆ BQ U SPACE ( s ( n )), which follows from the preceding theorem and a standard padding argument;for the sake of completeness, we now briefly state this argument.Suppose P = ( P , P ) ∈ BQSPACE ( s ( n )). By definition, there is some DSPACE ( s ( n ))-uniformfamily of general quantum circuits { Φ w = (Φ w, , . . . , Φ w,t w ) : w ∈ P } , where Φ w acts on h w = O ( s ( | w | )) qubits and has t w = 2 O ( s ( | w | )) gates, such that if w ∈ P , then Pr[Φ w accepts w ] ≥ , andif w ∈ P , then Pr[Φ w accepts w ] ≤ . Let M denote a deterministic Turing machine (DTM) thatproduces this family of circuits within the stated space bound. Let Σ denote the finite alphabetover which P is defined, and assume, without loss of generality, that { , } ⊆ Σ.We define P log = ( P log1 , P log0 ) ⊆ Σ ∗ such that P log j = { w s ( | w | ) : w ∈ P j } , for j ∈ { , } . We nextshow that P log ∈ BQL , by exhibiting a family of general quantum circuits { Φ log x = (Φ log x, , . . . , Φ log x,t log x ) : x ∈ P log } , with the appropriate parameters, that recognizes P log . Begin by noticing that, as s isspace-constructible, there is a DTM D that uses space O (log n ) on all inputs of length n , where, onany input x ∈ Σ ∗ , D checks if x = w s ( | w | ) , for some w ∈ Σ ∗ . If x is of this form, then D marks therightmost symbol of w ; otherwise, D rejects. We then construct a DTM M ′ which, on input x ∈ Σ ∗ produces Φ log x , as follows. First, M ′ runs D . If D rejects, then M ′ outputs a trivial single-gatecircuit, which acts on a single qubit and always rejects. Otherwise (i.e., when the input x is of theform w s ( | w | ) ), M ′ simulates M on the prefix w , producing the circuit Φ w ; note that, in this case, | x | = | w | + 1 + 2 s ( | w | ) , which implies M ′ runs in space O ( s ( | w | )) = O (log(2 s ( | w | ) )) = O (log( | x | )), andthat Φ log x acts on h log x = h w = O ( s ( | w | )) = O (log( | x | )) qubits and has t log x = t w = 2 O ( s ( | w | )) = | x | O (1) gates. Therefore, { Φ log x : x ∈ P log } is a L -uniform family of general quantum circuits, with theappropriate parameters, that recognizes P log with two-sided bounded-error .Thus, by Theorem 27, P log ∈ BQ U L . To complete the proof, we show that this implies P ∈ BQ U SPACE ( s ( n )). To see this, let { Q log x : x ∈ P log } denote a L -uniform family of (unitary) quantumcircuits that recognizes P log with two-sided bounded-error , where Q log x acts on O (log( | x | )) qubitsand has | x | O (1) gates, and let M U denote a logspace DTM that produces this circuit family. Wethen define a DTM M ′ U , which, on input w ∈ Σ ∗ , simply simulates M U on x = w s ( | w | ) ; inparticular, in order to keep track of the simulated head of M U when it is in the suffix 01 s ( | w | ) , M ′ U marks s ( | w | ) + 1 cells on its work tape (recall that s is space-constructible), which is then used asa classical counter that can count up to 2 s ( | w | )+1 −
1. By an analysis similar to that of the previousparagraph, we see that M ′ U runs in space O ( s ( n )) and that the circuit family { Q w : w ∈ P } that itproduces recognizes P and has the correct parameters.16 .3 Fully Logarithmic Approximation Schemes In this section, we study the class of functions that are well-approximable in quantum logspace,following (essentially) the notation and definitions of [16]. In particular, we work with the general(resp. unitary) quantum Turing machine model, rather than the equivalent model of a uniformfamily of general (resp. unitary) quantum circuits; of course, all results also apply to the quantumcircuit model. For simplicity, throughout this section, we fix the alphabet Σ = { , } . We say thata function f : Σ ∗ → R is poly -bounded if | f ( w ) | ≤ poly ( | w | ), ∀ w ∈ Σ ∗ . Definition 28.
We say that a poly -bounded f has a fully logarithmic quantum approximationscheme FLQAS if there is a (general) quantum TM M f that, on input h x, ǫ, δ i , where x ∈ Σ ∗ and ǫ, δ ∈ R > , runs in time poly ( | x | , ǫ − , log( δ − )) and space O (log( | x | ) + log( ǫ − ) + log(log( δ − ))),and outputs a value y ∈ R such that Pr[ | f ( x ) − y | ≥ ǫ ] ≤ δ (to be precise, M f outputs a stringthat encodes a dyadic rational number y ). In other words, with confidence at least 1 − δ , the value y is an additive approximation of f ( x ) with error at most ǫ . We analogously say that f has a FLQ U AS if M f is a unitary quantum TM, a FLRAS if M f is a randomized TM, and a
FLAS if M f is a deterministic TM (where, in this last case, we set δ = 0 and remove the dependence on δ fromthe time and space bounds).Following the notation established in Section 2.1 and Definitions 14 and 15, let D ( poly - matinv ) = [ n {h A, s, t i ∈ Σ ∗ : A ∈ d Mat( n ) , ≥ σ ( A ) ≥ σ n ( A ) ≥ n − O (1) , s, t ∈ { , . . . , n }} . In other words, D ( poly - matinv ) consists of precisely those strings in Σ ∗ that are encodings of in-stances of a variant of poly -conditioned- MATINV where only the portion of the promise involvingsingular values is required (i.e., there is no restriction on A − [ s, t ] involving b ). We then con-sider the poly -conditioned matrix inversion function poly - matinv : D ( poly - matinv ) → C , givenby poly - matinv ( h A, s, t i ) = A − [ s, t ]. For consistency with [16], as well as to make the rela-tionship between our results and certain logspace counting classes more clear, we then consider R ( poly - matinv ( · )) : D ( poly - matinv ) → R (resp. | poly - matinv ( · ) | : D ( poly - matinv ) → R ≥ ),which are given by the real part (resp. the magnitude) of the poly -conditioned matrix inversionfunction.Similarly, we define D ( poly - itmatprod ) ⊆ Σ ∗ to consist of all strings h A , . . . , A m , s, t i , where A , . . . , A m ∈ d Mat( n ) and s, t ∈ { , . . . , n } , where m = poly ( n ) and σ ( A j ,j ) ≤ poly ( n ) for1 ≤ j ≤ j ≤ m . We then define the function poly - itmatprod : D ( poly - itmatprod ) → C such that poly - itmatprod ( h A , . . . , A m , s, t i ) = A ,m [ s, t ]. We also define R ( poly - itmatprod ( · )) : D ( poly - itmatprod ) → R and | poly - itmatprod ( · ) | : D ( poly - itmatprod ) → R ≥ as above. Lastly,we define D ( poly - det ) = S n {h A i : A ∈ d Mat( n ) , ≥ σ ( A ) ≥ σ n ( A ) ≥ n − O (1) } and poly - det : D ( poly - det ) → C , such that, poly - det ( h A i ) = det( A ). Recall that the promise problem DET , givenin Definition 14, corresponds to approximating the function ln( | poly - det ( · ) | ) : D ( poly - det ) → R ≤ . Lemma 29. R ( poly - matinv ( · )) , | poly - matinv ( · ) | , R ( poly - itmatprod ( · )) , | poly - itmatprod ( · ) | , and ln( | poly - det ( · ) | ) each have a FLQ U AS .Proof. By [18, Theorem 14] (and the discussion following it), the functions R ( poly - matinv ( · )) and | poly - matinv ( · ) | each have a FLQ U AS (to be precise, the aforementioned theorem considered onlythe case in which A ∈ d Pos( n ), the general case then follows from the fact that the reductionfrom MATINV to MATINV + given by Lemma 23 preserves the value of the corresponding entryof the inverse matrix, not merely its magnitude); this improved upon the earlier result of Ta-Shma [45], which showed that these functions each have a FLQAS [16]. Notice that the reduction17rom
ITMATPROD to MATPOW given by Lemma 21, and the reduction from
MATPOW to MATINV given by Lemma 22, both also preserve the value of the matrix entry in question, not merely itsmagnitude. Therefore, R ( poly - itmatprod ( · )) and | poly - itmatprod ( · ) | each have a FLQ U AS . Finally,by Lemmas 16, 17 and 20, ln( | poly - det ( · ) | ) has a FLQ U AS .Note that, following [16], we have defined fully logarithmic (quantum, randomized, etc.) ap-proximation schemes with respect to additive error ǫ ; that is to say, we approximate f ( x ) by a value y such that Pr[ | f ( x ) − y | ≥ ǫ ] ≤ δ . We then define a multiplicative fully logarithmic (quantum,randomized, etc.) approximation scheme of a function g : Σ ∗ → R ≥ as an analogous procedurethat produces an approximation z such that Pr[ z [ e − ǫ g ( x ) , e ǫ g ( x )]] ≤ δ . Note that here, for con-venience, we follow the convention (as used in, for example, [21]) that multiplicative approximationsare defined using e ± ǫ , rather than the more standard (and essentially equivalent) (1 ± ǫ ). Lemma 30. ln( | poly - det ( · ) | ) has an (additive) FLQ U AS (resp. FLQAS , FLRAS , FLAS ) if and only if | poly - det ( · ) | has a multiplicative FLQ U AS (resp. FLQAS , FLRAS , FLAS ). In particular, | poly - det ( · ) | has a multiplicative FLQ U AS .Proof. The first statement follows immediately from the fact that | ln( | det( A ) | ) − y | ≥ ǫ if and onlyif e y [ e − ǫ | det( A ) | , e ǫ | det( A ) | ]. The second statement is a consequence of the first statement andLemma 29.Next, recall that by [16, Theorem 6], if BQL = BPL , then every poly -bounded function that hasa
FLQAS also has a
FLRAS (recall that we use
BQL and
BPL to denote classes of promise problems,which differs from the notation used in [16]). By combining this with the BQ U L -hardness of thevarious poly -conditioned promise problems (Theorem 4) and our result that BQ U L = BQL (Theo-rem 27), the following proposition is immediate; we note that a partial version of this propositionwas implicit in [15].
Proposition 31.
The following statements are equivalent.(i)
BQL = BPL .(ii) Every poly -bounded function that has a
FLQAS also has a
FLRAS .(iii) Every poly -bounded function that has a
FLQ U AS also has a FLRAS .(iv) R ( poly - matinv ( · )) has a FLRAS .(v) | poly - matinv ( · ) | has a FLRAS .(vi) R ( poly - itmatprod ( · )) has a FLRAS .(vii) | poly - itmatprod ( · ) | has a FLRAS .(viii) ln( | poly - det ( · ) | ) has a FLRAS .(ix) | poly - det ( · ) | has a multiplicative FLRAS .Remark.
In particular, the preceding proposition suggests that | poly - det ( · ) | does not have a multi-plicative FLRAS (as this would imply the seemingly unlikely statement
BQL = BPL ). It is naturalto compare this statement with the result of Jerrum, Sinclair, and Vigoda [21] which shows theexistence of a (multiplicative) FPRAS (fully polynomial randomized approximation scheme) for the permanent of a nonnegative integer matrix. 18
Well-Conditioned Singular
We next consider a well-conditioned version of the problem of determining if a matrix is singular.Additionally, following [41], we also consider well-conditioned versions of the “verification” versionsof standard
DET -complete problems.
Definition 32.
Consider functions m : N → N , κ : N → R ≥ , and ǫ : N → R > .(i) SINGULAR n,ǫ − Input : A ∈ \ Herm( n ). Promise : σ ( A ) ≤ σ n ( A ) ∈ { } ∪ [ ǫ ( n ) , Output : 1 if σ n ( A ) = 0, 0 otherwise.(ii) vMATPOW n,m,κ,ǫ − Input : A ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ Q [ i ] n . Promise : σ ( A j ) ≤ κ ( n ) , ∀ j ∈ { , . . . , m } , | b | ≤ κ ( n ), | A m [ s, t ] − b | ∈ { } ∪ [ ǫ ( n ) , κ ( n )]. Output : 1 if A m [ s, t ] = b , 0 otherwise.(iii) vITMATPROD n,m,κ,ǫ − Input : A , . . . , A m ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ Q [ i ] n . Promise : σ ( A j ,j ) ≤ κ ( n ) for 1 ≤ j ≤ j ≤ m , | b | ≤ κ ( n ), | A ,m [ s, t ] − b | ∈ { } ∪ [ ǫ ( n ) , κ ( n )]. Output : 1 if | A ,m [ s, t ] | = b , 0 otherwise.(iv) vMATINV n,κ,ǫ − Input : A ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ Q [ i ] n . Promise : σ ( A ) ≤ σ n ( A ) ≥ κ ( n ) , | b | ≤ κ ( n ), | A − [ s, t ] − b | ∈ { } ∪ [ ǫ ( n ) , κ ( n )]. Output : 1 if A − [ s, t ] = b , 0 otherwise. Remark.
Note that the requirement | b | ≤ κ ( n ) above is unnecessary due to the fact that, for anymatrix M , max s,t | M [ s, t ] | ≤ σ ( M ) (therefore, if | b | > κ ( n ), which can easily be checked, then theinput is a 0 instance). We include this condition only for convenience.We begin by exhibiting reductions between the above problems; in subsequent sections, we willuse these reductions to prove new properties of quantum logspace. Lemma 33. vITMATPROD ≤ m AC vMATPOW .Proof. Precisely analogous to the proof of Lemma 21.
Lemma 34. vMATPOW ≤ m AC vMATINV .Proof. Precisely analogous to the proof of Lemma 22.
Lemma 35. vMATINV ≤ m AC SINGULAR .Proof.
Consider h A, s, t, b i ∈ vMATINV n,κ,ǫ − . Let b B = (2 ⌈ κ ( n ) ⌉ A ) ⊕ (cid:16) − b ⌈ κ ( n ) ⌉ (cid:17) − I ∈ d Mat( n +1), u = χ s + χ n +1 , v = χ t + χ n +1 , and b C = b B − vu † ∈ d Mat( n + 1). By the matrix determinantlemma,det( b C ) = (1 − u b B − v ) det( b B ) = (cid:18) − (cid:18) A − [ s, t ]2 ⌈ κ ( n ) ⌉ + (cid:18) − b ⌈ κ ( n ) ⌉ (cid:19)(cid:19)(cid:19) det( b B ) = b − A − [ s, t ]2 ⌈ κ ( n ) ⌉ det( b B ) . A − [ s, t ] = b , then det( b C ) = 0, which implies σ n +1 ( b C ) = 0. Next, suppose | A − [ s, t ] − b | ≥ ǫ ( n ), then | det( b C ) | ≥ ǫ ( n )2 ⌈ κ ( n ) ⌉ | det( b B ) | . By the Weyl inequalities, σ ( b C ) ≤ σ ( b B ) + σ ( − vu † ) = σ ( b B ) + 1 and, for j ∈ { , . . . , n + 1 } , we have σ j ( b C ) ≤ σ j − ( b B ) + σ ( − vu † ) = σ j − ( b B ). Moreover, σ ( b B ) = 2 ⌈ κ ( n ) ⌉ σ ( A ) ≤ ⌈ κ ( n ) ⌉ , σ n ( b B ) = 2 ⌈ κ ( n ) ⌉ σ n ( A ) ≥
2, and σ n +1 ( b B ) = (cid:12)(cid:12)(cid:12) − b ⌈ κ ( n ) ⌉ (cid:12)(cid:12)(cid:12) − ≥ √ . Therefore, σ n +1 ( b C ) = | det( b C ) | σ ( b C ) n Q j =2 σ j ( b C ) ≥ ǫ ( n )2 ⌈ κ ( n ) ⌉ | det( b B ) | ( σ ( b B ) + 1) n − Q j =1 σ j ( b B ) = ǫ ( n ) σ n ( b B ) σ n +1 ( b B )2 ⌈ κ ( n ) ⌉ ( σ ( b B ) + 1) ≥ ǫ ( n ) √ ⌈ κ ( n ) ⌉ (2 ⌈ κ ( n ) ⌉ + 1) . Let b d = (2 ⌈ κ ( n ) ⌉ + 1) − and let b H = b d n +1 b C b C † n +1 ! ∈ \ Herm(2 n + 2). Notice that b H has eigenvalues {± b dσ ( b C ) , . . . , ± b dσ n +1 ( b C ) } . This implies σ ( b H ) = b dσ ( b C ) ≤ σ n +2 ( b H ) = b dσ n +1 ( b C ) ∈ { } ∪ h ǫ ( n ) √ ⌈ κ ( n ) ⌉ (2 ⌈ κ ( n ) ⌉ +1) , i . Moreover, σ n +2 ( b H ) = 0 ⇔ A − [ s, t ] = b . Therefore, h b H i ∈ SINGULAR n +2 , (2 ǫ ( n )) − √ ⌈ κ ( n ) ⌉ (2 ⌈ κ ( n ) ⌉ +1) and vMATINV ( h A, s, t, b i ) = SINGULAR ( h b H i ). U SPACE vs. RQMASPACE vs. QMASPACE For those promise problems P given in Definition 32, we define poly -conditioned- P as in Defini-tion 15. Lemma 36. poly -conditioned- vITMATPROD is coRQL -hard.Proof. Precisely analogous to the proof of Lemma 26.
Lemma 37.
RQMAL ⊆ RQ U L Proof.
Apply the well-known technique of replacing Merlin’s proof with the totally mixed state [30],which preserves perfect soundness [26]; then use space-efficient probability amplification for one-sided bounded-error (unitary) quantum logspace [50]. We briefly sketch the details.Suppose P = ( P , P ) ∈ RQMAL . By definition, there is a L -uniform family of (unitary) quantumcircuits { V w : w ∈ P } , where V w acts on m w + h w = O (log | w | ) qubits and has t w = poly ( | w | )gates, such that w ∈ P ⇒ ∃| ψ i ∈ Ψ m w , Pr[ V w accepts w, | ψ i ] ≥ c = , and w ∈ P ⇒ ∀| ψ i ∈ Ψ m w , Pr[ V w accepts w, | ψ i ] = k = 0, where Pr[ V w accepts w, | ψ i ] = k Π V w ( | ψ i ⊗ | h w i ) k .As in the proof of [30, Theorem 3.8], let A w = ( I mw ⊗ h h w | ) V † w Π V w ( I mw ⊗ | h w i ) ∈ Pos(2 m w );then w ∈ P ⇒ tr( A w ) ≥ c = and w ∈ P ⇒ tr( A w ) ≤ m w k = 0 ⇒ tr( A w ) = 0. Similar to theproof of [26, Theorem 14] (cf. [30, Theorem 3.10]), we define a L -uniform family of (unitary) quantumcircuits { Q w : w ∈ P } , where Q w acts on 2 m w + h w = O (log | w | ) qubits and has t w + O ( m w ) = poly ( | w | ) gates, such that, when Q w is applied to the state | m w + h w i , it simulates V w on | q i⊗ | h w i ,where | q i ∈ Ψ m w is drawn uniformly at random from the 2 m w standard basis elements of Ψ m w . Wehave Pr[ Q w accepts w ] = tr( A w − m w I mw ) = 2 − m w tr( A w ); thus, w ∈ P ⇒ Pr[ Q w accepts w ] = 2 − m w tr( A w ) ≥ − ( m w +1) = 1 /poly ( | w | ) , and w ∈ P ⇒ Pr[ Q w accepts w ] = 2 − m w tr( A w ) = 0 . Therefore, P ∈ Q U SPACE (log n ) poly ( n ) , . By [50, Lemma 5.1] Q U SPACE (log n ) poly ( n ) , = RQ U L ,which implies P ∈ RQ U L . 20 emma 38. poly -conditioned- SINGULAR is QMAL -hard.Proof. Suppose P = ( P , P ) ∈ QMAL . By definition, there is a L -uniform family of (unitary)quantum circuits { V w = ( V w, , . . . , V w,t w ) : w ∈ P } , where V w acts on m w + h w = O (log | w | ) qubitsand has t w = poly ( | w | ) gates, such that w ∈ P ⇒ ∃| ψ i ∈ Ψ m w , Pr[ V w accepts w, | ψ i ] ≥ c = 1, and w ∈ P ⇒ ∀| ψ i ∈ Ψ m w , Pr[ V w accepts w, | ψ i ] ≤ k = , where Pr[ V w accepts w, | ψ i ] = k Π V w ( | ψ i ⊗| h w i ) k .We make use of the Kitaev clock Hamiltonian construction [24, Section 14.4], in a manner similarto [18, Lemma 21] (though, without the need to first apply space-efficient probability amplificationtechniques). Let d w = 2 m w + h w ( t w + 1) = poly ( | w | ), define the d w -dimensional Hilbert space H w = C mw ⊗ C hw ⊗ C t w +1 , and let Π b = I b − ⊗ | ih | ⊗ I mw + hw − b ∈ d Proj(2 m w + h w ) denote theprojection onto the subspace of C mw ⊗ C hw spanned by states in which the b th qubit is 1. Wedefine the Hamiltonians H propw , H inw , H outw , H w ∈ d Pos( d w ) on H w as follows: H propw = 12 t w X j =1 (cid:16) − V w,j ⊗ | j ih j − | − V † w,j ⊗ | j − ih j | + I mw + hw ⊗ ( | j ih j | + | j − ih j − | ) (cid:17) H inw = m w + h w X b = m w +1 (Π b ⊗ | ih | ) , H outw = Π ⊗ | t w ih t w | , and H w = H inw + H propw + H outw . By [24, Section 14.4], ∃ r , r ∈ R > , such that σ ( H w ) ≤ r , ∀ w ∈ P , σ d w ( H w ) ≤ − ct w +1 = 0 , ∀ w ∈ P and σ d w ( H w ) ≥ r −√ kt w +1 = 1 /poly ( d w ) , ∀ w ∈ P . Therefore, h H w i ∈ poly -conditioned- SINGULAR and P ( w ) = SINGULAR ( h H w i ); as { V w : w ∈ P } is L -uniform, we see that H w may be constructedfrom w in L , which implies P ≤ m L poly -conditioned- SINGULAR . Lemma 39. poly -conditioned-
SINGULAR ∈ QMAL .Proof. Follows from using the quantum walk based Hamiltonian simulation technique of Childs[7, 11] to allow the phase estimation of [18, Lemma 19] to be carried out with one-sided error, inthe style of [18, Proposition 32], we omit the straightforward details.
Lemma 40. poly -conditioned-
SINGULAR is QMAL -complete.Proof. Follows immediately from Lemmas 38 and 39
Theorem 41.
RQMAL = RQ U L ⊆ RQL ⊆ coQMAL .Proof. Trivially, RQ U L ⊆ RQL and RQ U L ⊆ RQMAL . By Lemma 37,
RQMAL ⊆ RQ U L , whichimplies RQMAL = RQ U L . By Lemma 36, poly -conditioned- vITMATPROD is coRQL -hard; thus,by Lemmas 33 to 35, poly -conditioned- SINGULAR is coRQL -hard. Finally, by Lemma 39, wehave poly -conditioned- SINGULAR ∈ QMAL , which implies coRQL ⊆ QMAL ; therefore, RQL ⊆ coQMAL .We now prove Theorem 2, stated in the introduction, which we restate here for convenience. Theorem 2.
For any space-constructible function s : N → N , where s ( n ) = Ω(log n ) , we have RQMASPACE ( s ( n )) = RQ U SPACE ( s ( n )) ⊆ RQSPACE ( s ( n )) ⊆ coQMASPACE ( s ( n )) . Proof.
Follows from Theorem 41 and a padding argument analogous to that of Theorem 1.21 .2 NQSPACE vs. NQ U SPACE vs. NQMASPACE vs. PreciseQMASPACE We next consider variants of the well-conditioned problems of Definition 32, in which ǫ ( n ) = 0, ∀ n ∈ N . Definition 42.
Consider functions m : N → N and κ : N → R ≥ .(i) PreciseSINGULAR n Input : A ∈ \ Herm( n ). Promise : σ ( A ) ≤ σ n ( A ) ∈ [0 , Output : 1 if σ n ( A ) = 0, 0 otherwise.(ii) vPreciseMATPOW n,m,κ Input : A ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ Q [ i ] n . Promise : σ ( A j ) ≤ κ ( n ) , ∀ j ∈ { , . . . , m } , | b | ≤ κ ( n ), | A m [ s, t ] − b | ∈ [0 , κ ( n )]. Output : 1 if A m [ s, t ] = b , 0 otherwise.(iii) vITMATPROD n,m,κ Input : A , . . . , A m ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ Q [ i ] n . Promise : σ ( A j ,j ) ≤ κ ( n ) for 1 ≤ j ≤ j ≤ m , | b | ≤ κ ( n ), | A ,m [ s, t ] − b | ∈ [0 , κ ( n )]. Output : 1 if | A ,m [ s, t ] | = b , 0 otherwise.(iv) vPreciseMATINV n,κ Input : A ∈ d Mat( n ), s, t ∈ { , . . . , n } , b ∈ Q [ i ] n . Promise : σ ( A ) ≤ σ n ( A ) ≥ κ ( n ) , | b | ≤ κ ( n ), | A − [ s, t ] − b | ∈ [0 , κ ( n )]. Output : 1 if A − [ s, t ] = b , 0 otherwise.By arguments precisely analogous to those of Section 4.1, we obtain the following lemmas; weomit the proofs. Lemma 43. poly -conditioned- vPreciseITMATPROD is coNQL -hard. Lemma 44.
NQMAL ⊆ NQ U L Lemma 45.
PreciseSINGULAR is PreciseQMAL -complete. Theorem 46.
NQMAL = NQ U L = NQL = coPreciseQMAL = coC = L .Proof. Trivially, NQ U L ⊆ NQL and NQ U L ⊆ NQMAL . By Lemma 44,
NQMAL ⊆ NQ U L , which im-plies NQMAL = NQ U L . By Lemma 43, poly -conditioned- vPreciseITMATPROD is coNQL -hard; thus,by Lemmas 33 to 35, PreciseSINGULAR is coNQL -hard. By Lemma 45, poly -conditioned- SINGULAR ∈ PreciseQMAL , which implies NQL ⊆ coPreciseQMAL . By [5, Theorem 14], PreciseSINGULAR is C = L -complete, and by Lemma 45, PreciseSINGULAR is PreciseQMAL -complete; therefore, C = L = PreciseQMAL . Thus far, we have shown NQMAL = NQ U L ⊆ NQL ⊆ coPreciseQMAL = coC = L .To complete the proof, note that, by [49, Theorem 4.14], NQ U L = coC = L .We now prove Theorem 3, stated in the introduction, which we restate here for convenience. Theorem 3.
For any space-constructible function s : N → N , where s ( n ) = Ω(log n ) , we have NQMASPACE ( s ( n )) = NQ U SPACE ( s ( n )) = NQSPACE ( s ( n ))= coPreciseQMA SPACE ( s ( n )) = coC = SPACE ( s ( n )) . Proof.
Follows from Theorem 46 and a padding argument analogous to that of Theorem 1.22
Well-Conditioned Minimum Eigenvalue and Circuit Trace
In Section 3.1, a “scaled-down” version of the well-conditioned matrix inversion problem was shownto be BQ U L -complete, where the “standard” version of this problem is BQP -complete [20]. We nextconsider “scaled-down” versions of the minimum eigenvalue problem and of the unitary circuit traceestimation problem, the “standard” versions of which are
QMA -complete [24] and
DQC1 -complete[25, 42, 44], respectively. Interestingly, the “scaled-down” versions are both BQ U L -complete.Recall that DQC1 ⊆ BQP ⊆ QMA . It seems reasonable to suspect that both of the precedinginclusions are proper. However, the fact that the “scaled-down” versions of a
DQC1 -completeproblem, a
BQP -complete problem, and a
QMA -complete are all BQ U L -complete shows that thedifferences between the time-bounded classes (if these classes are, indeed, different) disappear inthe space-bounded setting.In the following, we say that a function f : D → [ − , D ⊆ R , is κ - Lipschitz if | f ( x ) − f ( y ) | ≤ κ | x − y | , ∀ x, y ∈ D . For some H ∈ Herm( n ), with eigendecomposition H = n P j =1 λ j ( H ) v j v † j , and for some function f : D → [ − , { λ j ( H ) : 1 ≤ j ≤ n } ⊆ D , let f ( H ) = n P j =1 f ( λ j ( H )) v j v † j ∈ Herm( n ). We use the term L -transducer to refer to a deterministic logspaceTuring machine that computes a function. As before, for some unitary circuit Q = ( Q , . . . , Q t ),where each Q j ∈ U( n ) (i.e., Q j is a unitary gate acting on log n qubits), we also use Q to denotethe element Q t · · · Q ∈ U( n ) that corresponds to applying the gates Q , . . . , Q t sequentially. Definition 47.
Consider functions κ : N → R ≥ and ǫ : N → R > .(i) MINEIGENVALUE + n,ǫ − Input : H ∈ d Pos( n ), b ∈ [0 , Promise : λ n ( H ) ∈ [0 , b ] ∪ [ b + ǫ ( n ) , Output : 1 if λ n ( H ) ≤ b , 0 otherwise.(ii) MINEIGENVALUE + ,gapn,ǫ − Input : H ∈ d Pos( n ), b ∈ [0 , Promise : λ n ( H ) ∈ [0 , b ] ∪ [ b + ǫ ( n ) , λ n − ( H ) ≥ b + ǫ ( n ), λ n − ( H ) − λ n ( H ) ≥ ǫ ( n ). Output : 1 if λ n ( H ) ≤ b , 0 otherwise.(iii) CIRCUITTRACE n,ǫ − Input : Q = ( Q , . . . , Q t ), where each Q j ∈ U( n ), b ∈ [ − , Promise : n tr( Q ) ∈ [ − , b − ǫ ( n )] ∪ [ b, Output : 1 if n tr( Q ) ≥ b , 0 otherwise.(iv) FILTEREDMATTRACE n,κ,ǫ − Input : H ∈ \ Herm( n ), b ∈ [ − , M f a L -transducer that computes f : D → [ − , Promise : { λ j ( H ) : 1 ≤ j ≤ n } ⊆ D , f is κ ( n )-Lipschitz, n tr( f ( H )) ∈ [ − , b − ǫ ( n )] ∪ [ b, Output : 1 if n tr( f ( H )) ≥ b , 0 otherwise.(v) MATINV + , ≥ n,κ,ǫ − Input : H ∈ d Pos( n ), s, t ∈ { , . . . , n } , b ∈ R ≥ ǫ ( n ) . Promise : σ ( H ) ≤ σ n ( H ) ≥ κ ( n ) , H − [ s, t ] ∈ [0 , b − ǫ ( n )] ∪ [ b, κ ( n )]. Output : 1 if H − [ s, t ] ≥ b , 0 otherwise. 23vi) SUMMATINV ≥ n,κ,ǫ − Input : H ∈ \ Herm( n ), S ⊆ { , . . . , n } , b ∈ R ≥ ǫ ( n ) . Promise : σ ( H ) ≤ σ n ( H ) ≥ κ ( n ) , det( H ) > | S | P ( s,t ) ∈ S H − [ s, t ] ∈ [0 , b − ǫ ( n )] ∪ [ b, κ ( n )]. Output : 1 if | S | P ( s,t ) ∈ S H − [ s, t ] ≥ b , 0 otherwise.Fefferman and Lin [18] showed that the poly -conditioned- MINEIGENVALUE + problem and the poly -conditioned- MATINV + , ≥ problem are both BQ U L -complete. We next show, directly, that MATINV + , ≥ ≤ m AC MINEIGENVALUE + ,gap . This statement (combined with the fact that, triv-ially, MINEIGENVALUE + ,gap ≤ m AC MINEIGENVALUE + ) provides alternate proofs of the BQ U L -hardness of poly -conditioned- MINEIGENVALUE + and of poly -conditioned- MATINV + , ≥ ∈ BQ U L .In particular, the previous proof of the BQ U L -hardness of poly -conditioned- MINEIGENVALUE + [18,Lemma 21] required the use of “strong” in-place error reduction for BQ U L [18, Lemma 28], whereas poly -conditioned- MATINV + , ≥ was shown to be BQ U L -hard by a much simpler argument that didnot require the use of any such error reduction machinery [18, Theorem 13]. As a consequence ofour direct proof of the fact that MATINV + , ≥ ≤ m AC MINEIGENVALUE + ,gap , we then obtain a simpleproof of the BQ U L -hardness of poly -conditioned- MINEIGENVALUE + .We then study CIRCUITTRACE and
FILTEREDMATTRACE . We show that the poly -conditionedversions of both problems are BQ U L -complete. In particular, we show MINEIGENVALUE + ,gap ≤ m AC FILTEREDMATTRACE , which, combined with the above results, provides an extremely simple proofof the fact that poly -conditioned-
MINEIGENVALUE + , gap , poly -conditioned- MATINV + , ≥ ∈ BQ U L .Note that FILTEREDMATTRACE is a “scaled-down” version of a problem that was shown to be in
DQC1 by Cade and Montanaro [10]; their technique has many common elements with the techniqueused earlier by Harrow, Hassidim, and Lloyd [20] in their proof of the fact that “scaled-up” matrixinversion is in
BQP . We use the term “filtered” in reference to the “filter functions” that appearin these proofs.
Lemma 48.
MATINV + , ≥ ≤ m AC SUMMATINV ≥ .Proof. Consider h H, s, t, b i ∈
MATINV + , ≥ n,κ,ǫ − . First, suppose s = t ; then h H, { s } , b i ∈ SUMMATINV ≥ n,κ,ǫ and MATINV ( h H, s, s, b i ) = SUMMATINV ( h H, { s } , b i ). Next, suppose instead that s = t . If n is even, let b A = H and m = n ; if n is odd, let b A = H ⊕ I and m = n + 1. Let b H = m b A † b A m ! ∈ \ Herm(2 m ), b S = { s, t } , and b b = b . Then h b H, b S, b b i ∈ SUMMATINV ≥ m,κ ( n ) ,ǫ − ( n ) and MATINV ( h H, s, t, b i ) = SUMMATINV ( h b H, b S, b b i ). Lemma 49.
SUMMATINV ≥ ≤ m AC MINEIGENVALUE + ,gap .Proof. Consider h H, S, b i ∈
SUMMATINV ≥ n,κ,ǫ − . Let v = √ | S | P s ∈ S χ s , b C = b − ǫ ( n ) vv † ∈ \ Herm( n ), b A = H − b C ∈ \ Herm( n ), and x = v † H − v = | S | P ( s,t ) ∈ S H − [ s, t ]. By the matrix determinantlemma, det( b A ) = (1 − x b − ǫ ( n ) ). If x ≥ b , then det( b A ) ≤ − ǫ ( n )2 κ ( n ) det( H ); if x ≤ b − ǫ ( n ), thendet( b A ) ≥ ǫ ( n )2 κ ( n ) det( H ).By the Weyl eigenvalue inequalities, for any j ∈ { , . . . , n − } , λ j ( b A ) ≥ λ j +1 ( H ) − λ ( b C ) =24 j +1 ( H ) > λ j ( b A ) ≤ λ j ( H ) + λ ( − b C ) = λ j ( H ) ≤
1. This implies,det( H ) κ ( n ) ≥ det( H ) λ − n ( H ) = n − Y j =1 λ j ( H ) ≥ n − Y j =1 λ j ( b A ) ≥ n − Y j =1 λ j +1 ( H ) = det( H ) λ − ( H ) ≥ det( H ) > . Clearly, det( b A ) = n Q j =1 λ j ( b A ) = λ n ( b A ) n − Q j =1 λ j ( b A ). Therefore, if det( b A ) ≤ − ǫ ( n )2 κ ( n ) det( H ), then λ n ( b A ) ≤ − ǫ ( n )2 κ ( n ) ; similarly, if det( b A ) ≥ ǫ ( n )2 κ ( n ) det( H ), then λ n ( b A ) ≥ ǫ ( n )2 κ ( n ) . By the promise, b ≥ ǫ ( n ), whichimplies λ n ( b A ) ≥ λ n ( H ) + λ n ( − b C ) = λ n ( H ) − b − ǫ ( n ) ≥ −
1. Moreover, λ n ( b A ) ≤ λ n ( H ) ≤ b H = ( I n +1 + ( b A ⊕ )) ∈ d Pos( n + 1), where 0 denotes the 1 × λ n +1 ( b H ) ∈ h , (cid:16) − ǫ ( n )2 κ ( n ) (cid:17)i ∪ (cid:8) (cid:9) , λ n ( H ) − λ n +1 ( H ) ≥ ǫ ( n )2 κ ( n ) , and λ n +1 ( b H ) ∈ h , (cid:16) − ǫ ( n )2 κ ( n ) (cid:17)i ⇔ x ≥ b . Let b b = (cid:16) − ǫ ( n )2 κ ( n ) (cid:17) . Then h b H, b b i ∈ MINEIGENVALUE + ,gapn +1 , κ ( n ) ǫ − ( n ) and SUMMATINV ( h H, S, b i ) = MINEIGENVALUE ( h b H , b b i ). Lemma 50. poly -conditioned-
MINEIGENVALUE + ,gap is BQ U L -complete.Proof. By [18, Theorem 13], poly -conditioned-
MATINV + , ≥ is BQ U L -hard, which, by Lemmas 48and 49, implies poly -conditioned- MINEIGENVALUE + ,gap is BQ U L -hard. By [18, Lemmas 19 and 20], poly -conditioned- MINEIGENVALUE + ∈ BQ U L ; trivially, MINEIGENVALUE + ,gap ≤ m AC MINEIGENVALUE + ,which implies poly -conditioned- MINEIGENVALUE + ,gap ∈ BQ U L . Lemma 51. poly -conditioned-
CIRCUITTRACE is BQ U L -complete.Proof. Follows straightforwardly from the same technique used in the “standard” proof that the“scaled-up” circuit trace estimation problem is
DQC1 -complete [25, 42, 44]; we omit the details.
Lemma 52.
MINEIGENVALUE + ,gap ≤ m AC FILTEREDMATTRACE
Proof.
Consider h H, b i ∈
MINEIGENVALUE + ,gapn,ǫ − . Let b D = [0 , b ] ∪ [ b + ǫ ( n ) ,
1] and let b f : b D → { , } be defined such that b f ( x ) = 1 if x ≤ b , and b f ( x ) = 0 if x ≥ b + ǫ ( n ). Clearly, { λ j ( H ) : 1 ≤ j ≤ n } ⊆ b D and b f is ǫ − ( n )-Lipschitz. Moreover,tr( b f ( H )) = n X j =1 b f ( λ j ( H )) = |{ j : λ j ( H ) ≤ b }| = ( , λ n ( H ) ≤ b , otherwise.Let b b = n . Then n tr( b f ( H )) ∈ { , b b } and n tr( b f ( H )) = b b ⇔ λ n ( H ) ≤ b . Thus, h b H, b b, M b f i ∈ FILTEREDMATTRACE n,ǫ − ,ǫ − and MINEIGENVALUE ( h H, b i ) = FILTEREDMATTRACE ( h b H , b b, M b f i ). Lemma 53. poly -conditioned-
FILTEREDMATTRACE is BQ U L -complete.Proof. By Lemmas 50 and 52, poly -conditioned-
FILTEREDMATTRACE is BQ U L -hard. Moreover, poly -conditioned- FILTEREDMATTRACE ∈ BQ U L follows, straightforwardly, from the same tech-nique used to prove [10, Lemma 1] (which shows that a “scaled-up” version of this problem is in DQC1 ); we omit the details. 25
Discussion
We conclude by stating a few interesting open problems related to our work. In Theorem 1 weestablished the equivalence of unitary quantum space, general quantum space, and space-boundedquantum Merlin-Arthur proof systems, in the two-sided bounded-error case. We obtained an anal-ogous equivalence for one-sided unbounded-error in Theorem 3. However, in the case of one-sidedbounded-error, we only have the partial results of Theorem 2. In particular, specializing to the caseof logspace, we have
BQL = BQ U L = QMAL in the two-sided bounded-error case (Theorem 27), andwe have
RQMAL = RQ U L ⊆ RQL ⊆ coQMAL in the one-sided bounded-error case (Theorem 41).It is naturally to ask if the analogues of results known to hold for two-sided bounded-error alsohold for one-sided bounded-error. Open Problem 1. Is RQ U L = RQL ? Is
RQL = coQMAL ?By the well-known result of Zachos and F¨urer [55], MA = MA ; that is to say, it is possibleto achieve perfect completeness for classical (polynomial time) Merlin-Arthur proof systems. Onthe other hand, the question of whether or not it is possible to achieve perfect completeness for quantum (polynomial time) Merlin-Arthur proof systems (i.e., is QMA = QMA ?) remains open(see, for instance, [1, 3, 9, 22] for previous discussion). We next consider the logspace analogue ofthis question. Open Problem 2. Is QMAL = QMAL ?A possible explanation for the difficulty of proving QMA = QMA (if these classes are in-deed equal) was provided by Aaronson’s result [1] that there is a quantum oracle U such that QMA U = QMA U ; therefore, any proof of QMA = QMA must use a technique that is quantumlynonrelativizing . Note that the technique used by Zachos and F¨urer [55] to show MA = MA is (clas-sically) relativizing. It is not hard to see that Aaronson’s argument can also be used to produce aquantum oracle U such that QMAL U = QMAL U , and so any proof of QMAL = QMAL must alsouse quantumly nonrelativizing techniques. We emphasize that the techniques used in this paperto show our results concerning new inclusions between various complexity classes (i.e., the variousreductions between linear-algebraic problems shown in this paper) are quantumly nonrelativizing .Moreover, it is known that it is possible to achieve perfect completeness in quantum Merlin-Arthur proof systems that have a classical witness ; that is to say, QCMA = QCMA [22]. Note that,trivially, BQ U L ⊆ QCMAL ⊆ QMAL . Thus, the known equality BQ U L = QMAL immediately implies
QCMAL = QMAL . Therefore,
QMAL = QMAL ⇔ QCMAL = QMAL ⇐ QCMAL = QCMAL .Returning to an issue discussed in Section 1.1, Boix-Adser`a, Eldar, and Mehraban [8] showed κ ( n )-conditioned- DET ∈ DSPACE (log( n ) log( κ ( n )) poly (log log n )). Moreover, they asked the ques-tion “is poly -conditioned- DET BQL -complete?” In this paper, we answered their question in theaffirmative. Therefore, if
BQL DSPACE (log − ǫ n ), ∀ ǫ >
0, then both our result and the re-sult of Boix-Adser`a, Eldar, and Mehraban are essentially optimal (in terms of the relationshipbetween condition number and needed space). Recall that, by the classic result of Watrous [51],
BQL ⊆ DSPACE (log n ).Moreover, we note that our paper and that of Boix-Adser`a, Eldar, and Mehraban used similarpower series techniques to produce space-efficient algorithms for κ ( n )-conditioned- DET . However,our quantum algorithm can make use of a power series with an exponentially larger number ofterms than seems possible for their (or any other) classical algorithm. This suggests a possiblemechanism for explaining the supposed advantage of quantum computers over classical computersin the space-bounded setting. We conclude with a general question.26 pen Problem 3.
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A A QTM-based Proof of BQL=BQ U L In this appendix, we show that the technique that was used to show
BPL ⊆ BQ U L , in Proposition 25,also immediately shows BQL ⊆ BQ U L (and, therefore, BQL = BQ U L ). Here, BQL is defined in termsof a logspace quantum Turing machine (QTM), as was the case in, for instance [23,31,35,45,49–52],rather than the equivalent model of an L -uniform family of general quantum circuits used in thispaper.For concreteness, we use the classically controlled logspace (general) QTM defined by Wa-trous [51] (with the minor alteration that we require all transition amplitudes of the QTM to becomputable in L ); however, we note that our result would apply equally well to any “reasonable”logspace QTM model that is classically controlled (this includes all models considered in all of thepapers cited above). In brief, such a QTM M consists of a (classical) finite control, an internalquantum register of constant size, a classical “measurement” register of constant size, and threetapes: (1) a read-only input tape that, on any input w , contains the string L w R , where L and R are special symbols that serve as left and right end-markers, (2) a read/write classical29ork tape consisting of s ( | w | ) = O (log | w | ) cells, each of which holds a symbol from some finitealphabet Γ, and (3) a read/write quantum work tape, consisting of s ( | w | ) = O (log | w | ) qubits. Eachof the tapes has a single bidirectional head. At the start of the computation, both work-tapes are“blank” (to be precise, each cell of the classical work tape contains some specified blank-symbol inΓ and each qubit of the quantum work tape is in the state | i ); each qubit of the internal quantumregister is also in the state | i . Each step of the computation of M involves applying a selectivequantum operation to the combined register consisting of the internal quantum register and thesingle qubit that is currently under the head of the quantum work tape; the particular choice ofwhich selective quantum operation to perform may depend on the state of the finite control and thesymbols currently under the heads of the input tape and classical work tape. The (classical) resultof this quantum operation is stored in the measurement register. Then, depending on this result,as well as on the state of the finite control and the symbols currently under the heads of the inputtape and classical work tape, the classical configuration of the machine evolves; to be precise, thestate of the finite control is updated, a symbol is written on the classical work-tape, and the headof each work tape moves up to one cell in either direction. The machine accepts (resp.) rejectsits input by entering a special (classical) accepting (resp. rejecting state). See [51] for a completedefinition. Proposition 54.
BQL = BQ U L .Proof. Trivially,