Complexity limitations on one-turn quantum refereed games
CComplexity limitations on one-turnquantum refereed games
Soumik Ghosh John Watrous
Institute for Quantum Computing and School of Computer ScienceUniversity of Waterloo, Canada
February 4, 2020
Abstract
This paper studies complexity theoretic aspects of quantum refereed games, whichare abstract games between two competing players that send quantum states to areferee, who performs an efficiently implementable joint measurement on the twostates to determine which of the player wins. The complexity class QRG ( ) containsthose decision problems for which one of the players can always win with high prob-ability on yes-instances and the other player can always win with high probabilityon no-instances, regardless of the opposing player’s strategy. This class trivially con-tains QMA ∪ co-QMA and is known to be contained in PSPACE. We prove strongercontainments on two restricted variants of this class. Specifically, if one of the play-ers is limited to sending a classical (probabilistic) state rather than a quantum state,the resulting complexity class CQRG ( ) is contained in ∃ · PP (the nondeterministicpolynomial-time operator applied to PP); while if both players send quantum statesbut the referee is forced to measure one of the states first, and incorporates the classi-cal outcome of this measurement into a measurement of the second state, the resultingclass MQRG ( ) is contained in P · PP (the unbounded-error probabilistic polynomial-time operator applied to PP).
1. Introduction
Abstract notions of games have long played an important role in complexity theory. Forexample, combinatorial games provide complete problems for various complexity classes[DH09], the notion of alternation is naturally described in game-theoretic terms [CKS81],and interactive proof systems [Bab85, BM88, GMR85, GMR89] and many variants of themare naturally formulated as games [Con87, FKS95].1 a r X i v : . [ c s . CC ] F e b his paper is concerned with games between two competing, computationally un-bounded players, administered by a computationally bounded referee. In the classicalsetting, complexity theoretic aspects of games of this form were investigated in the 1990sby Koller and Megiddo [KM92], Feigenbaum, Koller, and Shor [FKS95], Condon, Feigen-baum, Lund, and Shor [CFLS95, CFLS97], and Feige and Kilian [FK97]. Quantum compu-tational analogues of these games were later considered in [GW05], [Gut05], [GW07], and[JW09].Our focus will be on one-turn refereed games , in which the players and the referee firstreceive a common input string, and then each player sends a single polynomial-length(quantum or classical) message to the referee, who then decides which player has won. Wewill refer to the two competing players as Alice and
Bob for convenience. In the classicalcase Alice and Bob’s messages may in general be described by probability distributionsover strings, while in the quantum case Alice and Bob’s messages are described by mixedquantum states, which are represented by density operators. In both cases, the referee’sdecision process must be specified by a polynomial-time generated family of (quantumor classical) circuits. Two complexity classes are defined—RG ( ) in the classical setting and QRG ( ) in the quantum setting—consisting of all promise problems A = ( A yes , A no ) for which there exists a game (either classical or quantum, respectively) such that Alicecan win with high probability on inputs x ∈ A yes and Bob can win with high probabilityon inputs x ∈ A no , regardless of the other player’s behavior.In essence, the complexity classes RG ( ) and QRG ( ) may be viewed as extensions ofthe classes MA and QMA in which two competing Merlins , one whose aim is to convincethe referee (whose role is analogous to Arthur, also called the verifier, in the case of MAand QMA) that the input string is a yes-instance of a given problem, and the other whoseaim is to convince the referee that the input string is a no-instance.It is known that the complexity class RG ( ) is equal to S p , which refers to the secondlevel of the symmetric polynomial-time hierarchy introduced by Canetti [Can96] and Russelland Sundaram [RS98]. This class is most typically defined in terms of quantifiers thatsuggest games in which Alice and Bob choose polynomial-length strings (as opposed toprobability distributions of strings) to send to the referee, but the class does not change ifone adopts a bounded-error definition in which Alice and Bob are allowed to make use ofrandomness [FIKU08]. Moreover, the class does not change if the referee is permitted theuse of randomness, again assuming a bounded-error definition. An essential fact throughwhich these equivalence may be proved, due to Althöfer [Alt94] and Lipton and Young[LY94], is that non-interactive randomized games always admit near-optimal strategiesthat are uniform over polynomial-size sets of strings. It is also known that RG ( ) is closedunder Cook reductions [RS98] and satisfies RG ( ) ⊆ ZPP NP [Cai07].In contrast to the containment RG ( ) ⊆ ZPP NP , the best upper-bound known forQRG ( ) is that this class is contained in PSPACE [JW09]. It is reasonable to conjecture We note explicitly that this nomenclature clashes with [FK97], which defines RG ( ) in terms of one-round (i.e., two-turn) refereed games, which is RG ( ) with respect to our naming conventions. ( ) can be proved. Indeed, Gutoski and Wu [GW13]proved that QRG ( ) = PSPACE, where QRG ( ) is a two-turn analogue of QRG ( ) , inwhich the referee first sends polynomial-length quantum messages to Alice and Bob,then receives responses from them, and finally decides which player wins. The classicalanalogue of QRG ( ) , which we denote by RG ( ) , is also known to be equal to PSPACE[FK97].In this work we consider two restricted variants of QRG ( ) , and prove stronger upper-bounds than PSPACE on these restricted variants. The first variant is one in which Alice islimited to sending a classical message to the referee, while Bob is free to send a quantumstate. The resulting class, which we call CQRG ( ) , is proved to be contained in ∃ · PP (theclass obtained when the nondeterministic polynomial-time operator is applied to PP).This containment follows from an application of the Althöfer–Lipton–Young techniquementioned above, although in the quantum setting the proof requires relatively recenttail bounds on sums of matrix-valued random variables, as opposed to a more standardHoeffding–Chernoff type of bound that suffices in the classical case. In particular, wemake use of a tail bound of this sort due to Tropp [Tro12]. The second variant we consideris one in which both Alice and Bob are free to send quantum states, but where the refereemust first measure Alice’s state and then incorporate the classical outcome of this mea-surement into a measurement of Bob’s state. We call the corresponding class MQRG ( ) ,and prove the containment MQRG ( ) ⊆ P · PP (the class obtained when the unboundederror probabilistic polynomial-time operator is applied to PP).
2. Preliminaries
We assume the reader is familiar with basic aspects of computational complexity theoryand quantum information and computation. There are four subsections included in thispreliminaries section, the first of which clarifies a few specific concepts, conventions, anddefinitions concerning complexity theory. The second subsection is concerned specificallywith counting complexity, and presents a development of some results on this topic thatare central to this paper. Proofs are included because these results represent minor gen-eralizations of known results on counting complexity. The third subsection discusses afew specific definitions and concepts from quantum information and computation, alongwith a proof of a fact that may be considered a known result, but for which a completeproof does not appear in published form. The final subsection states the tail bound dueto Tropp mentioned above.
Complexity theory basics
Throughout this paper, languages, promise problems, and functions on strings are as-sumed to be over the binary alphabet Σ = {
0, 1 } . The set of natural numbers, including 0,is denoted N . 3 function of the form p : N → N is said to be polynomially bounded if there exists adeterministic Turing machine that runs in polynomial time and outputs 0 p ( n ) on input 0 n for all n ∈ N . Unless it is explicitly indicated otherwise, the input of a given polynomiallybounded function p is assumed to be the natural number | x | , for whatever input string x ∈ Σ ∗ is being considered at that moment. With this understanding in mind, we willwrite p in place of p ( | x | ) when referring to the natural number output that is determinedin this way. For example, in Definition 1 below, all of the occurrences of p in the displayedequations are short for p ( | x | ) . This convention helps to make formulas and equationsmore clear and less cluttered.A promise problem is a pair A = ( A yes , A no ) of sets of strings A yes , A no ⊆ Σ ∗ with A yes ∩ A no = ∅ . Strings in A yes represent yes-instances of a decision problem, strings in A no represent no-instances, and all other strings represent “don’t care” inputs for whichno restrictions are placed on a hypothetical computation for that problem.We fix a pairing function that efficiently encodes two strings x , y ∈ Σ ∗ into a singlebinary string denoted (cid:104) x , y (cid:105) ∈ Σ ∗ , and we assume that this function satisfies some simpleproperties:1. The length of the pair (cid:104) x , y (cid:105) depends only on the lengths | x | and | y | , and is polynomialin these lengths.2. The computation of x and y from (cid:104) x , y (cid:105) , as well as the computation of (cid:104) x , y (cid:105) from x and y , can be performed deterministically in polynomial time.One suitable choice for such a function is suggested by the equation (cid:104) a a · · · a n , b b · · · b m (cid:105) = a a · · · a n b b · · · b m (1)for a , a , . . . , a n , b , b , . . . , b m ∈ Σ . Any such pairing function may be extended recur-sively to obtain a tuple function for any fixed number of inputs by taking (cid:104) x , x , x , . . . , x k (cid:105) = (cid:104)(cid:104) x , x (cid:105) , x , . . . , x k (cid:105) (2)for strings x , . . . , x k ∈ Σ ∗ , where k ≥
3. Hereafter, when we refer to the computation ofany function taking multiple string-valued arguments, we assume that these input stringshave been encoded into a single string using this tuple function. For instance, when f is afunction that represents a computation, we write f ( x , y , z ) rather than f ( (cid:104) x , y , z (cid:105) ) .Finally, we define the nondeterministic and probabilistic polynomial-time operators,which may be applied to an arbitrary complexity class, as follows. Definition 1.
For a given complexity class of languages C , the complexity classes ∃ · C andP · C are defined as follows.1. The complexity class ∃ · C contains all promise problems A = ( A yes , A no ) for whichthere exists a language B ∈ C and a polynomially bounded function p such that thesetwo implications hold: x ∈ A yes ⇒ (cid:110) y ∈ Σ p : (cid:104) x , y (cid:105) ∈ B (cid:111) (cid:54) = ∅ , x ∈ A no ⇒ (cid:110) y ∈ Σ p : (cid:104) x , y (cid:105) ∈ B (cid:111) = ∅ . (3)4. The complexity class P · C contains all promise problems A = ( A yes , A no ) for whichthere exists a language B ∈ C and a polynomially bounded function p such that thesetwo implications hold: x ∈ A yes ⇒ (cid:12)(cid:12)(cid:12)(cid:110) y ∈ Σ p : (cid:104) x , y (cid:105) ∈ B (cid:111)(cid:12)(cid:12)(cid:12) > · p , x ∈ A no ⇒ (cid:12)(cid:12)(cid:12)(cid:110) y ∈ Σ p : (cid:104) x , y (cid:105) ∈ B (cid:111)(cid:12)(cid:12)(cid:12) ≤ · p . (4) Counting complexity
Counting complexity is principally concerned with the number of solutions to certaincomputational problems. Readers interested in learning more about counting complexityand some of its applications are referred to the survey paper of Fortnow [For97]. As wassuggested at the beginning of the current section, we will require some basic results oncounting complexity that represent minor generalizations of known results. We beginwith the following definition.
Definition 2.
Let C be any complexity class of languages over the alphabet Σ . A function f : Σ ∗ → Z is a Gap · C function if there exist languages A , B ∈ C and a polynomiallybounded function p such that f ( x ) = (cid:12)(cid:12)(cid:8) y ∈ Σ p : (cid:104) x , y (cid:105) ∈ A (cid:9)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:8) y ∈ Σ p : (cid:104) x , y (cid:105) ∈ B (cid:9)(cid:12)(cid:12) (5)for all x ∈ Σ ∗ .We observe that this definition is slightly non-standard, as gap functions are usuallydefined in terms of differences between the number of accepting and rejecting computa-tions of nondeterministic machines (as opposed to a difference involving two potentiallyunrelated languages A and B ). It is also typical that one focuses on specific choices for C ,particularly C = P. Our definition is, however, equivalent to the traditional definition inthis case, and we will write GapP rather than Gap · P so as to be consistent with the stan-dard name for this class of functions. We will also be interested in the case C = PP, whichyields a class of functions Gap · PP that is less commonly considered.The following proposition is immediate from the definitions of Gap · C and P · C . Proposition 3.
Let C be a complexity class of languages that is closed under complementation.A promise problem A = ( A yes , A no ) is contained in P · C if and only if there exists a Gap · C function f such that x ∈ A yes ⇒ f ( x ) > x ∈ A no ⇒ f ( x ) ≤
0. (6)A key feature of the class of GapP functions that facilitates its use is that it possessstrong closure properties. This is true more generally for the class Gap · C provided that C itself possesses certain properties. For the closure properties we require, it suffices that C
5s nontrivial (meaning that C contains at least one language that is not equal to ∅ or Σ ∗ )and is closed under the join operation as well as polynomial-time truth-table reductions.(The join of languages A and B is defined as { x : x ∈ A } ∪ { x : x ∈ B } .) Theseproperties are, of course, possessed by both P and PP, with the closure of PP under truthtable reductions having been proved by Fortnow and Reingold [FR96] based on methodsdeveloped by Beigel, Reingold, and Spielman [BRS95].The lemmas that follow establish the specific closure properties we require. For thefirst property the assumption that C is closed under joins and polynomial-time truth-tablereductions is not required; closure under Karp reductions suffices. Lemma 4.
Let C be a nontrivial complexity class of languages that is closed under Karp reduc-tions. Let f ∈ Gap · C and let p be a polynomially bounded function. The functiong ( x ) = ∑ y ∈ Σ p f ( x , y ) (7) is a Gap · C function.Proof. By the assumption that f ∈ Gap · C , there exists a polynomially bounded function q and languages A , A ∈ C such that f ( x , y ) = (cid:12)(cid:12)(cid:8) z ∈ Σ q ( |(cid:104) x , y (cid:105)| ) : (cid:104) x , y , z (cid:105) ∈ A (cid:9)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:8) z ∈ Σ q ( |(cid:104) x , y (cid:105)| ) : (cid:104) x , y , z (cid:105) ∈ A (cid:9)(cid:12)(cid:12) (8)for all x ∈ Σ ∗ and y ∈ Σ p . By the assumptions on our pairing function described above, itis the case that |(cid:104) x , y (cid:105)| depends only on | x | and | y | , and therefore there exists a (necessarilypolynomially bounded) function r such that r ( | x | ) = p ( | x | ) + q ( |(cid:104) x , y (cid:105)| ) for all x ∈ Σ ∗ and y ∈ Σ p . Define B = (cid:8) (cid:104) x , yz (cid:105) : y ∈ Σ p , z ∈ Σ q ( |(cid:104) x , y (cid:105)| ) , (cid:104) x , y , z (cid:105) ∈ A (cid:9) , B = (cid:8) ( x , yz ) : y ∈ Σ p , z ∈ Σ q ( |(cid:104) x , y (cid:105)| ) , (cid:104) x , y , z (cid:105) ∈ A (cid:9) . (9)By the nontriviality and closure of C under Karp reductions, it is evident that B , B ∈ C .It may be verified that g ( x ) = (cid:12)(cid:12)(cid:8) w ∈ Σ r : (cid:104) x , w (cid:105) ∈ B (cid:9)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:8) w ∈ Σ r : (cid:104) x , w (cid:105) ∈ B (cid:9)(cid:12)(cid:12) (10)for all x ∈ Σ ∗ , and therefore g ∈ Gap · C .For the next lemma, and elsewhere in the paper, we will use the following notation forconvenience: Σ n denotes the set of all strings over the binary alphabet Σ that have length n and contain exactly one occurrence of the symbol 1. It is therefore the case that | Σ n | = n . Lemma 5.
Let C be a nontrivial complexity class of languages that is closed under joins andpolynomial-time truth table reductions. Let f ∈ Gap · C and let p be a polynomially boundedfunction. The function g ( x ) = ∏ y ∈ Σ p f ( x , y ) (11) is a Gap · C function. roof. Given that f ∈ Gap · C , there exists a polynomially bounded function q and lan-guages A , A ∈ C such that f ( x , y ) = (cid:12)(cid:12)(cid:12)(cid:110) z ∈ Σ q ( | ( x , y ) | ) : (cid:104) x , y , z (cid:105) ∈ A (cid:111)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:110) z ∈ Σ q ( | ( x , y ) | ) : (cid:104) x , y , z (cid:105) ∈ A (cid:111)(cid:12)(cid:12)(cid:12) (12)for all x , y ∈ Σ ∗ . We may assume further that A and A are disjoint languages, for if theyare not, we may replace A and A with A ∩ A and A ∩ A , respectively; this does notchange the value of the right-hand side of the equation (12), and the languages A ∩ A and A ∩ A must both be contained in C for A , A ∈ C by the closure of C under joinsand truth-table reductions.By the assumptions on our pairing function described above, there exists a polynomi-ally bounded function r such that r ( | x | ) = q ( | ( x , y ) | ) for all x ∈ Σ ∗ and y ∈ Σ p . We willwrite y , . . . , y p to denote the elements of Σ p sorted in lexicographic order. Define twolanguages B and B as follows: • B is the language of all pairs (cid:104) x , z · · · z p (cid:105) , where x ∈ Σ ∗ and z , . . . , z p ∈ Σ r , forwhich there exists a string w ∈ Σ p having even parity such that (cid:104) x , y , z (cid:105) ∈ A w , . . . , (cid:104) x , y p , z p (cid:105) ∈ A w p . (13) • B is the language of all pairs (cid:104) x , z · · · z p (cid:105) , where x ∈ Σ ∗ and z , . . . , z p ∈ Σ r , forwhich there exists a string w ∈ Σ p having odd parity such that (cid:104) x , y , z (cid:105) ∈ A w , . . . , (cid:104) x , y p , z p (cid:105) ∈ A w p . (14)Given that A and A are disjoint and contained in C , along with the fact that C is closedunder joins and truth-table reductions, it follows that B , B ∈ C . The lemma now followsfrom the observation that g ( x ) = (cid:12)(cid:12)(cid:12)(cid:110) z ∈ Σ s : (cid:104) x , z (cid:105) ∈ B (cid:111)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:110) z ∈ Σ s : (cid:104) x , z (cid:105) ∈ B (cid:111)(cid:12)(cid:12)(cid:12) (15)for all x ∈ Σ ∗ , where s = p · r . Lemma 6.
Let C be a nontrivial complexity class of languages that is closed under joins andpolynomial-time truth table reductions, let f , f ∈ Gap · C , and let p and q be polynomiallybounded functions. For every string x ∈ Σ ∗ and y ∈ Σ q , define the matrix M x , y as Re (cid:0) (cid:104) z | M x , y | w (cid:105) (cid:1) = f ( x , y , z , w ) ,Im (cid:0) (cid:104) z | M x , y | w (cid:105) (cid:1) = f ( x , y , z , w ) , (16) for all z , w ∈ Σ p . There exist Gap · C functions g and g satisfying Re (cid:0) (cid:104) z | M x , y · · · M x , y q | w (cid:105) (cid:1) = g ( x , z , w ) ,Im (cid:0) (cid:104) z | M x , y · · · M x , y q | w (cid:105) (cid:1) = g ( x , z , w ) , (17) for all x ∈ Σ ∗ and z , w ∈ Σ p , where y , . . . , y q denote the elements of Σ q sorted in lexicographicorder. roof. By the assumptions on C stated in the lemma, there must exist a Gap · C function h satisfying h ( x , y , 0 z , 0 w ) = f ( x , y , z , w ) , h ( x , y , 0 z , 1 w ) = f ( x , y , z , w ) , h ( x , y , 1 z , 0 w ) = − f ( x , y , z , w ) , h ( x , y , 1 z , 1 w ) = f ( x , y , z , w ) , (18)for all x ∈ Σ ∗ , y ∈ Σ q , and z , w ∈ Σ p . The matrix N x , y defined as (cid:104) u | N x , y | v (cid:105) = h ( x , y , u , v ) (19)for all x ∈ Σ ∗ , y ∈ Σ q and u , v ∈ Σ p + may be visualized as a 2 × N x , y = (cid:32) Re ( M x , y ) Im ( M x , y ) − Im ( M x , y ) Re ( M x , y ) (cid:33) . (20)We observe that N x , y · · · N x , y q = (cid:32) Re (cid:0) M x , y · · · M x , y q (cid:1) Im (cid:0) M x , y · · · M x , y q (cid:1) − Im (cid:0) M x , y · · · M x , y q (cid:1) Re (cid:0) M x , y · · · M x , y q (cid:1)(cid:33) . (21)Given that h is a Gap · C function, there must exist a Gap · C function F for which F ( x , u · · · u q , y k ) = h ( x , y k , u k − , u k ) (22)for all x ∈ Σ ∗ , u , . . . , u q ∈ Σ p + , and k ∈ {
1, . . . , q } .Finally, define G ( x , u · · · u q ) = ∏ y ∈ Σ q F ( x , u · · · u q , y ) = h ( x , y , u , u ) · · · h ( x , y q , u q − , u q ) (23)for all x ∈ Σ ∗ and u , . . . , u q ∈ Σ p + , as well as g ( x , z , w ) = ∑ u ∈ Σ ( q − )( p + ) G ( x , 0 zu w ) , g ( x , z , w ) = ∑ u ∈ Σ ( q − )( p + ) G ( x , 0 zu w ) , (24)for all x ∈ Σ ∗ and z , w ∈ Σ p . It follows by Lemmas 4 and 5 that g , g ∈ Gap · C .Observing that g and g satisfy the equations (17), which is perhaps most evidentfrom the equation (21), the proof of the lemma is complete.8 uantum information and quantum circuits The notation we use when discussing quantum information is standard for the subject,and we refer the reader to the books [NC00, KSV02, Wil17, Wat18] for further details. Acouple of points concerning quantum information notation and conventions that may behelpful to some readers follow.First, when we refer to a register X , we mean a collection of qubits that we wish to viewas a single entity, and we then use the same letter X in a scripted font to denote the finite-dimensional complex Hilbert space associated with X (i.e., the space of complex vectorshaving entries indexed by binary strings of length equal to the number of qubits in X ).The set of density operators acting on such a space is denoted D ( X ) .Second, a channel transforming a register X into a register Y is a completely positiveand trace-preserving linear map Φ that transforms each density operator ρ ∈ D ( X ) into adensity operator Φ ( ρ ) ∈ D ( Y ) . (More generally, such a mapping Φ transforms arbitrarylinear operators acting on X into linear operators acting on Y .) The adjoint of such a chan-nel Φ is the uniquely determined linear map Φ ∗ transforming linear operators acting on Y into linear operators acting on X that satisfies the equationTr (cid:0) P Φ ( ρ ) (cid:1) = Tr (cid:0) Φ ∗ ( P ) ρ (cid:1) (25)for all density operators ρ ∈ D ( X ) and all positive semidefinite operators P acting on Y .The adjoint Φ ∗ of a channel Φ is not necessarily itself a channel, but rather is a completelypositive and unital linear map, which means that Φ ∗ ( Y ) = X (for X and Y denot-ing the identity operators acting on X and Y , respectively). Intuitively speaking, if P isa measurement operator in the equation above, one can think of Φ ∗ as transforming themeasurement operator P into a new measurement operator Φ ∗ ( P ) , with the probability ofthis outcome for the state ρ being the same as if one first applied Φ to ρ and then measuredwith respect to P .Now we will move on to quantum circuits , which are acyclic networks of quantumgates connected by qubit wires. We choose to use the standard, general model of quan-tum information based on density operators and quantum channels, as opposed to therestricted model of pure state vectors and unitary operations, when discussing quantumcircuits. In this general model, each gate represents a quantum channel acting on a con-stant number of qubits—including nonunitary gates, such as gates that introduce freshinitialized qubits or gates that discard qubits. Through this model, ordinary classical cir-cuits, as well as classical circuits that introduce randomness into computations, can beviewed as special cases of quantum circuits. One may also represent measurements di-rectly as quantum gates or circuits.It is well-known that this general model is equivalent to the purely unitary model, asis explained in [AKN98] and [Wat09], for instance. The main benefits of using the generalmodel in the context of this paper are that (i) it allows us to avoid having to constantly dis-tinguish between input qubits and ancillary qubits, or output qubits and garbage qubits,and (ii) it has the minor but nevertheless positive side effect of eliminating the appearance9f the irrational number 1/ √ Hadamard , Toffoli , and phase-shift gates (whichinduce the single-qubit unitary transformation determined by the actions | (cid:105) (cid:55)→ | (cid:105) and | (cid:105) (cid:55)→ i | (cid:105) ), as well as ancillary gates and erasure gates . Ancillary gates take no input qubitsand output a single qubit in the | (cid:105) state, while erasure gates take one input qubit andproduce no output qubits, and are described by the partial trace. Any other choice for theunitary gates that is universal for quantum computing could be taken instead, but thegate set just specified is both simple and convenient.The size of a quantum circuit is defined to be the number of gates in the circuit plus thetotal number of input and output qubits. Thus, if a quantum circuit were to be representedin a standard way as a directed acyclic graph, its size would simply be the number ofvertices, including a vertex for each input and output qubit, of the corresponding graph.A collection { Q x : x ∈ Σ } of quantum circuits is said to be polynomial-time generated if there exists a polynomial-time deterministic Turing machine that, on input x ∈ Σ ∗ ,outputs an encoding of the circuit Q x . When such a family is parameterized by tuples ofstrings, it is to be understood that we are implicitly referring to one of the tuple-functionsdiscussed previously. We will not have any need to discuss the specifics of the encodingscheme that we use, but naturally it is assumed to be efficient, with the size of a circuitand its encoding length being polynomially related.The following lemma relates the complexity of computing circuit transition ampli-tudes to GapP functions. The essential idea it expresses is due to Fortnow and Rogers[FR99], who proved a variant of it for unitary computations by quantum Turing machines.While a result along the lines of the lemma that follows is suggested in the survey paper[Wat09], that paper does not include a proof, and so we include one below. Lemma 7.
Let { Q x : x ∈ Σ ∗ } be a polynomial-time generated family of quantum circuits, whereeach circuit Q x takes n input qubits and outputs k qubits, for polynomially bounded functions nand k. There exists a polynomially bounded function r and GapP functions f and g such that Re (cid:0) (cid:104) u | Q x (cid:0) | z (cid:105)(cid:104) w | (cid:1) | v (cid:105) (cid:1) = − r f ( x , z , w , u , v ) ,Im (cid:0) (cid:104) u | Q x (cid:0) | z (cid:105)(cid:104) w | (cid:1) | v (cid:105) (cid:1) = − r f ( x , z , w , u , v ) , (26) for all x ∈ Σ ∗ , z , w ∈ Σ n , and u , v ∈ Σ k .Proof. Consider first an arbitrary channel Φ that maps n -qubit density operators to k -qubit density operators. The action of Φ on density operators is linear, and can thereforebe represented through matrix multiplication. One concrete way to do this is to use theso-called natural representation (also known as the linear representation) of quantumchannels.A description of the natural representation of a quantum channel begins with the vectorization mapping: assuming M is a matrix whose rows and columns are indexed by10trings of some length m , the corresponding vector vec ( M ) is indexed by strings of length2 m according to the following definition:vec ( M ) = ∑ y , z ∈ Σ m (cid:104) y | M | z (cid:105) | yz (cid:105) . (27)In words, the vectorization map reshapes a matrix into a vector by transposing the rowsof the matrix into column vectors and stacking them on top of one another.With respect to the vectorization mapping, the action of the channel Φ is described byits natural representation K ( Φ ) , which is a linear mapping that acts as K ( Φ ) vec ( ρ ) = vec ( Φ ( ρ )) (28)for every n -qubit density operator ρ . As a matrix, K ( Φ ) has columns indexed by stringsof length 2 n and rows indexed by strings of length 2 k . Its entries are described explicitlyby the equation (cid:104) uv | K ( Φ ) | zw (cid:105) = (cid:104) u | Φ ( | z (cid:105)(cid:104) w | ) | v (cid:105) (29)holding for every z , w ∈ Σ n and u , v ∈ Σ k . The equations (26) may therefore be equiva-lently written as Re (cid:0) (cid:104) uv | K ( Q x ) | zw (cid:105) (cid:1) = − r f ( x , z , w , u , v ) ,Im (cid:0) (cid:104) uv | K ( Q x ) | zw (cid:105) (cid:1) = − r f ( x , z , w , u , v ) . (30)It must be observed that the natural representation is multiplicative, in the sense thatchannel composition corresponds to matrix multiplication: K ( ΦΨ ) = K ( Φ ) K ( Ψ ) for allchannels Φ and Ψ for which the composition ΦΨ makes sense. It is also helpful to notethat a channel Φ ( ρ ) = U ρ U ∗ corresponding to a unitary operation has as its naturalrepresentation the operator K ( Φ ) = U ⊗ U . (31)Now let us turn to the family { Q x : x ∈ Σ ∗ } . Because this family is polynomial-timegenerated, there must exist a polynomially bounded function r for which size ( Q x ) ≤ r for all x ∈ Σ ∗ . We may therefore write Q x = Q x , r · · · Q x ,1 (32)for Q x ,1 , . . . , Q x , r being either identity channels or channels that describe the action ofa single gate of Q x tensored with the identity channel on all of the qubits besides theinputs of the corresponding gate that exist at the moment that the gate is applied. We alsoobserve that the number of input qubits and output qubits of each Q x , k must be boundedby r .Given that K ( Q x ) = K ( Q x , r ) · · · K ( Q x ,1 ) , (33)we are led to consider the natural representation of each channel Q x , k . It will be conve-nient to identify each operator K ( Q x , k ) with the matrix indexed by strings of length 2 r ,11s opposed to being indexed by strings whose lengths depend on the number of qubitsin existence before and after Q x , k is applied, simply by padding K ( Q x , k ) with rows andcolumns of zero entries.The natural representations of the individual gates in the universal gate set we haveselected are as follows:1. Hadamard gate: 12 − −
11 1 − − − − (34)2. Phase gate: − i i
00 0 0 1 (35)3. Toffoli gate: ⊗ (36)4. Ancillary qubit gate: (37)5. Erasure gate: (cid:0) (cid:1) (38)Based on these representations, it is straightforward to define GapP functions (or, in fact,FP functions) g and g such thatRe (cid:0) (cid:104) uv | K ( Q x , k ) | zw (cid:105) (cid:1) = g ( x , z , w , u , v , y k ) ,Im (cid:0) (cid:104) uv | K ( Q x , k ) | zw (cid:105) (cid:1) = g ( x , z , w , u , v , y k ) , (39)12or all x ∈ Σ ∗ , k ∈ {
1, . . . , r } , and u , v , z , w ∈ Σ r , where we write y , . . . , y r to denote theelements of Σ r sorted in lexicographic order. It now follows through a straightforwardapplication of Lemma 6 there must exist GapP functions f and f satisfying (26) andtherefore (30), for all x ∈ Σ ∗ , z , w ∈ Σ n , and u , v ∈ Σ k , as required. A tail bound for operator-valued random variables
We will make use of the following tail bound on the minimum eigenvalue of the averageof a collection of operator-valued random variables. This bound follows from a moregeneral result due to Tropp. In particular, the bound stated in the theorem below followsfrom Theorem 5.1 of [Tro12] together with Pinsker’s inequality, which relates the relativeentropy of two distributions to their total variation distance.
Theorem 8 (Tropp) . Let d and N be positive integers, let η ∈ [
0, 1 ] and ε > be real numbers,and let X , . . . , X N be independent and identically distributed operator-valued random variableshaving the following properties:1. Each X k takes d × d positive semidefinite operator values satisfying X k ≤ .2. The minimum eigenvalue of the expected operator E ( X k ) satisfies λ min ( E ( X k )) ≥ η .It is the case that Pr (cid:18) λ min (cid:18) X + · · · + X N N (cid:19) < η − ε (cid:19) ≤ d exp ( − N ε ) . (40)
3. Complexity classes for one-turn quantum refereed games
In this section we define the complexity classes to be considered in this paper: QRG ( ) ,CQRG ( ) , and MQRG ( ) . The definitions of these classes all refer to the notion of a referee ,which (in this paper) is a polynomial-time generated family R = { R x : x ∈ Σ ∗ } (41)of quantum circuits having the following special form.1. For each x ∈ Σ ∗ , the inputs to the circuit R x are grouped into two registers: an n qubitregister A and an m -qubit register B , for polynomially bounded functions n and m .2. The output of each circuit R x is a single qubit, which is to be measured with respectto the standard basis immediately after the circuit is run.Given that classical probabilistic states may be viewed as special cases of quantum states(corresponding to diagonal density operators), this definition of a referee can still be usedin the situation in which either or both of the registers A and B is constrained to initiallystore a classical state. 13e are interested in the situation that, for a given choice of an input string x ∈ Σ ∗ , theinput to the circuit R x is a product state of the form ρ ⊗ σ , where ρ ∈ D ( A ) is a state ofthe register A and σ ∈ D ( B ) is a state of the register B . The state ρ ∈ D ( A ) is to be viewedas representing the state that Alice plays, while σ ∈ D ( B ) represents the state Bob plays.When the single output qubit of the circuit R x is measured with respect to the standardbasis, the outcome 1 is interpreted as “Alice wins,” while the outcome 0 is interpreted as“Bob wins.”Now, consider the quantity defined as ω ( R x ) = max ρ ∈ D ( A ) min σ ∈ D ( B ) (cid:104) | R x ( ρ ⊗ σ ) | (cid:105) . (42)Given that D ( A ) and D ( B ) are compact and convex sets, and the value (cid:104) | R x ( ρ ⊗ σ ) | (cid:105) isbilinear in ρ and σ , Sion’s min-max theorem implies that changing the order of the min-imum and maximum does not change the value of the expression. That is, this quantitymay alternatively be written ω ( R x ) = min σ ∈ D ( B ) max ρ ∈ D ( A ) (cid:104) | R x ( ρ ⊗ σ ) | (cid:105) . (43)This value represents the probability that Alice wins the game defined by the circuit R x ,assuming both Alice and Bob play optimally. With that definitions in hand, we may nowdefine the complexity class QRG ( ) , which is short for one-turn quantum refereed games . Definition 9.
A promise problem A = ( A yes , A no ) is contained in the complexity classQRG ( ) α , β if there exists a referee R = { R x : x ∈ Σ ∗ } such that the following propertiesare satisfied:1. For every string x ∈ A yes , it is the case that ω ( R x ) ≥ α .2. For every string x ∈ A no , it is the case that ω ( R x ) ≤ β .We also define QRG ( ) = QRG ( ) .In this definition, α and β may be constants, or they may be functions of the length of theinput x . A short summary of known facts and observations concerning the complexityclass QRG ( ) follows. • QMA ⊆ QRG ( ) . This is because the referee’s measurement may simply ignore Bob’sstate σ and treat Alice’s state ρ as a quantum proof in a QMA proof system. • QRG ( ) is closed under complementation: QRG ( ) = co-QRG ( ) . For a promiseproblem ( A yes , A no ) ∈ QRG ( ) , one may obtain a one-turn quantum refereed gamefor ( A no , A yes ) by simply exchanging the roles of Alice and Bob.14 It is the case that QRG ( ) = QRG ( ) α , β for a wide range of choices of α and β , sim-ilar to error bounds for BPP, BQP, and QMA. In particular, QRG ( ) = QRG ( ) α , β provided that α and β are polynomial-time computable and satisfy α ≤ − − p , β ≥ − p , and α − β ≥ p (44)for some choice of a strictly positive polynomially bounded function p . • QRG ( ) ⊆ PSPACE [JW09].The question that originally motivated the work reported in this paper is whetherthe containment QRG ( ) ⊆ PSPACE can be improved. We do not succeed in improvingthis containment, but we are able to prove stronger bounds on two interesting restrictedvariants of QRG ( ) , which we now define.The first variant of QRG ( ) we define is one in which Alice’s state is restricted to be aclassical state. We will call this class CQRG ( ) . Definition 10.
A promise problem A = ( A yes , A no ) is contained in the complexity classCQRG ( ) α , β if there exists a referee R = { R x : x ∈ Σ ∗ } such that the following propertiesare satisfied:1. For every string x ∈ Σ ∗ , the circuit R x takes the form illustrated in Figure 1. That is, R x takes an n -qubit register A and an m -qubit register B as input, measures each qubitof A with respect to the standard basis, leaving it in a classical state, and then runsthe circuit Q x on the pair ( A , B ) , producing a single output qubit.2. For every string x ∈ A yes , it is the case that ω ( R x ) ≥ α .3. For every string x ∈ A no , it is the case that ω ( R x ) ≤ β .We also define CQRG ( ) = CQRG ( ) .Formally speaking, the standard basis measurement suggested by Definition 10 canbe implemented by independently performing the completely dephasing channel on eachqubit of A . This channel can be constructed using the universal gate set we have selectedusing a Toffoli gate with suitably initialized inputs as follows: | (cid:105)| (cid:105) TrTr Error reduction may be performed through parallel repetition followed by majority vote. An analysisof this method for QRG(1) requires that one considers the possibility that the dishonest player (meaningthe one that should not have a strategy that wins with high probability) entangles his or her state acrossthe different repetitions, with the claimed bounds following from a similar analysis to parallel repetitionfollowed by majority vote for QMA [KSV02]. We note that there is no “in place” error reduction methodknown for QRG(1) that is analogous to the technique of [MW05] for QMA. x Q x AB Figure 1: A CQRG ( ) referee. The register A is initially measured (or, equivalently, de-phased) with respect to the standard basis, causing a classical state to be input into Q x ,along with the register B , which is unaffected by this standard basis measurement.Here the square labeled | (cid:105) is an ancillary gate, the square labeled | (cid:105) denotes an ancillarygate composed with a not-gate X = HPPH (for H and P denoting Hadamard and phase-shift gates), and the square labeled Tr denotes an erasure gate.In effect, a referee R that satisfies the first requirement of Definition 10 forces the stateAlice plays to be a classical state (i.e., a state represented by a diagonal density operator).That is, for any density operator ρ that Alice might choose to play, the state of A that isinput into Q x takes the form ∑ y ∈ Σ n p ( y ) | y (cid:105)(cid:104) y | (45)for some probability vector p over n -bit strings, and therefore the state that is pluggedinto the top n qubits of the circuit Q x represents a classical state. Given that the standardbasis measurement acts trivially on all diagonal states, we observe that Alice may causean arbitrary diagonal density operator of the form (45) to be input into Q x . In short, theset of possible states that may be input into the top n qubits of the circuit Q x is preciselythe set of diagonal n -qubit density operators.The second variant of QRG ( ) we define is one in which Alice and Bob both sendquantum states to the referee, but the referee first measures Alice’s state, obtaining a clas-sical outcome, which is then measured together with Bob’s state (as illustrated in Fig-ure 2). Definition 11.
A promise problem A = ( A yes , A no ) is contained in the complexity classMQRG ( ) α , β if there exists a referee R = { R x : x ∈ Σ ∗ } such that the following propertiesare satisfied:1. For every string x ∈ Σ ∗ , the circuit R x takes the form illustrated in Figure 2. That is, R x takes an n -qubit register A and an m -qubit register B as input, and first applies aquantum circuit P x to A , yielding a k -qubit register Y , for k a polynomially boundedfunction. The register Y is then measured with respect to the standard basis, so that it16 x Q x P x ρσ Figure 2: An MQRG ( ) referee.then contains a classical state, and finally a quantum circuit Q x is applied to the pair ( Y , B ) , yielding a single qubit.2. For every string x ∈ A yes , it is the case that ω ( R x ) ≥ α .3. For every string x ∈ A no , it is the case that ω ( R x ) ≤ β .We also define MQRG ( ) = MQRG ( ) .In essence, an MQRG ( ) referee measures Alice’s qubits with respect to a general,efficiently implementable measurement, which yields a k -bit classical outcome, which isthen plugged into Q x along with Bob’s quantum state.It is of course immediate thatCQRG(1) ⊆ MQRG(1) ⊆ QRG(1); (46)a CQRG(1) referee is a special case of an MQRG(1) referee in which P x is the identitymap on n qubits, while an MQRG(1) referee is a special case of a QRG(1) referee. We alsoobserve that both CQRG ( ) and MQRG ( ) are robust with respect to error bounds in thesame way as was described above for QRG ( ) .
4. Upper-bound on CQRG(1)
In this section, we prove that CQRG(1) is contained in ∃ ·
PP. The proof represents a fairlydirect application of the Althöfer–Lipton–Young [Alt94, LY94] technique, although (aswas suggested above) the quantum setting places a new demand on this technique thatrequires the use of a tail bound on sums of matrix-valued random variables. We willsplit the proof of this containment into two lemmas, followed by a short proof of themain theorem—this is done primarily because the lemmas will also be useful for provingMQRG(1) ⊆ P · PP in the section following this one. Some readers may wish to skip to thestatement and proof of Theorem 14 below, as it explains the purpose of these two lemmaswithin the context of that theorem. 17he first lemma represents an implication of Theorem 8 due to Tropp to the setting athand.
Lemma 12.
Let k and m be positive integers, let p ∈ P ( Σ k ) be a probability distribution onk-bit strings, let S y be a m × m positive semidefinite operator satisfying ≤ S y ≤ for eachy ∈ Σ k , and let N ≥ ( m + ) . For strings y , . . . , y N ∈ Σ k sampled independently from thedistribution p, it is the case that Pr (cid:32) λ min (cid:32) S y + · · · + S y N N (cid:33) < λ min (cid:32) ∑ y ∈ Σ k p ( y ) S y (cid:33) − (cid:33) <
13 . (47)
Proof.
Define X , . . . , X N to be independent and identically distributed operator-valuedrandom variables, each taking the (operator) value S y with probability p ( y ) , for every y ∈ Σ k . The expected value of each of these random variables is therefore given by P = ∑ y ∈ Σ k p ( y ) S y . (48)By taking η = λ min ( P ) and ε = (cid:18) λ min (cid:18) X + · · · + X N N (cid:19) < λ min ( P ) − (cid:19) ≤ m exp (cid:18) − N (cid:19) <
13 , (49)which is equivalent to the bound stated in the lemma.The second lemma uses counting complexity to relate the minimum eigenvalue ofmeasurement operators defined by quantum circuits to PP languages. We note that thetechnique of weakly estimating the largest eigenvalue of a measurement operator usingthe trace of a power of that operator, through the relations (53) appearing in the proofbelow, is the essential idea behind the unpublished proof of the containment QMA ⊆ PPclaimed in [KW00].
Lemma 13.
Let { Q x : x ∈ Σ ∗ } be a polynomial-time generated family of quantum circuits,where each circuit Q x takes as input a k-qubit register Y and an m-qubit register B , for polynomi-ally bounded functions k and m, and outputs a single qubit. For each x ∈ Σ ∗ and y ∈ Σ k , definean operator S x , y = (cid:0) (cid:104) y | ⊗ B (cid:1) Q ∗ x ( | (cid:105)(cid:104) | ) (cid:0) | y (cid:105) ⊗ B (cid:1) . (50) For every polynomially bounded function N, there exists a language B ∈ PP for which the follow-ing implications are true for all x ∈ Σ ∗ and y , . . . , y N ∈ Σ k : λ min (cid:18) S x , y + · · · + S x , y N N (cid:19) ≥ ⇒ ( x , y · · · y N ) ∈ B , (51) λ min (cid:18) S x , y + · · · + S x , y N N (cid:19) ≤ ⇒ ( x , y · · · y N ) (cid:54)∈ B . (52)18 roof. The essence of the proof is that if P is an operator whose entries have real andimaginary parts proportional to GapP functions, and r is a polynomially bounded func-tion, then there exists a GapP function that is proportional to the real part of Tr ( P r ) . When P is a 2 m × m positive semidefinite operator, this allows one to choose r to be sufficientlylarge, but still polynomially bounded, so that a GapP function is obtained that takes posi-tive or negative values in accordance with the required implications (51) and (52), throughthe use of the following bounds relating the largest eigenvalue and the trace of any such P : λ max ( P ) r = λ max ( P r ) ≤ Tr ( P r ) ≤ m λ max ( P r ) = m λ max ( P ) r . (53)In the case at hand, it will suffice to take r = m .In greater detail, let us begin by defining T x , y = (cid:0) (cid:104) y | ⊗ B (cid:1) Q ∗ x ( | (cid:105)(cid:104) | ) (cid:0) | y (cid:105) ⊗ B (cid:1) (54)for each x ∈ Σ ∗ and y ∈ Σ k . Observe that S x , y and T x , y are positive semidefinite operatorssatisfying S x , y + T x , y = B , so that the implication in the statement of the lemma mayalternatively be written as λ max (cid:18) T x , y + · · · + T x , y N N (cid:19) ≤ ⇒ ( x , y · · · y N ) ∈ B , (55) λ max (cid:18) T x , y + · · · + T x , y N N (cid:19) ≥ ⇒ ( x , y · · · y N ) (cid:54)∈ B . (56)Thus, if the operator P x , y ··· y N = T x , y + · · · + T x , y N N (57)satisfies λ max ( P x , y ··· y N ) ≤ ( P mx , y ··· y N ) ≤ m m < m (58)while if λ max ( P x , y ··· y N ) ≥ ( P mx , y ··· y N ) ≥ (cid:16) (cid:17) m > m . (59)By Lemma 7 there exists a polynomially bounded function r along with GapP func-tions f and g satisfyingRe ( (cid:104) z | T x , y | w (cid:105) ) = Re (cid:0) (cid:104) | Q x (cid:0) | yz (cid:105)(cid:104) yw | (cid:1) | (cid:105) (cid:1) = − r f ( x , y , z , w ) ,Im ( (cid:104) z | T x , y | w (cid:105) ) = − Im (cid:0) (cid:104) | Q x (cid:0) | yz (cid:105)(cid:104) yw | (cid:1) | (cid:105) (cid:1) = − r g ( x , y , z , w ) , (60)for all x ∈ Σ ∗ , y ∈ Σ k , and z , w ∈ Σ m . Define functions F and G as follows: F ( x , y · · · y N , z , w ) = f ( x , y , z , w ) + · · · + f ( x , y N , z , w ) , G ( x , y · · · y N , z , w ) = g ( x , y , z , w ) + · · · + g ( x , y N , z , w ) , (61)19or all x ∈ Σ ∗ , y , . . . , y N ∈ Σ k , and z , w ∈ Σ m . It is the case that F and G are GapPfunctions satisfying F ( x , y · · · y N , z , w ) = r · N · Re ( (cid:104) z | P x , y ··· y N | w (cid:105) ) , G ( x , y · · · y N , z , w ) = r · N · Im ( (cid:104) z | P x , y ··· y N | w (cid:105) ) . (62)Through an application of Lemmas 4 and 6, we conclude that there must exist a GapPfunction H satisfying H ( x , y · · · y N ) = rm · N m · Tr (cid:0) P mx , y ··· y N (cid:1) . (63)The GapP function K ( x , y · · · y N ) = rm · N m − m · H ( x , y · · · y N ) (64)therefore takes positive values if λ max ( P x , y ··· y N ) ≤ λ max ( P x , y ··· y N ) ≥ B as claimed. Theorem 14.
CQRG ( ) ⊆ ∃ · PP .Proof. Let A = ( A yes , A no ) be any promise problem contained in CQRG ( ) , let a refereebe fixed that establishes the inclusion A ∈ CQRG ( ) , and let { Q x : x ∈ Σ ∗ } be thecollection of circuits that describes this referee, in accordance with Definition 10.Let x ∈ A yes ∪ A no be any input string. Consider first the situation that Alice plays de-terministically, sending a string y ∈ Σ n to the referee, so that ρ = | y (cid:105)(cid:104) y | . Having selected astate ρ representing Alice’s play, we are effectively left with a binary-valued measurementbeing performed on the state sent to the referee by Bob. We observe that, for any choice ofa state σ ∈ D ( B ) representing Bob’s play, the probabilities that the referee’s measurementgenerates the outcomes 0 and 1 are given by (cid:104) | Q x ( | y (cid:105)(cid:104) y | ⊗ σ ) | (cid:105) and (cid:104) | Q x ( | y (cid:105)(cid:104) y | ⊗ σ ) | (cid:105) , (65)respectively. By defining an operator S x , y ∈ Pos ( B ) as S x , y = (cid:0) (cid:104) y | ⊗ B (cid:1) Q ∗ x ( | (cid:105)(cid:104) | ) (cid:0) | y (cid:105) ⊗ B (cid:1) , (66)we therefore obtain the measurement operator corresponding to the 1 outcome of thismeasurement, as Tr (cid:0) S x , y σ (cid:1) = (cid:104) | Q x ( | y (cid:105)(cid:104) y | ⊗ σ ) | (cid:105) (67)and Tr (cid:0) ( B − S x , y ) σ (cid:1) = (cid:104) | Q x ( | y (cid:105)(cid:104) y | ⊗ σ ) | (cid:105) (68)for all σ ∈ D ( B ) .Now, as Bob aims to minimize the probability for outcome 1 to appear, the relevantproperty of the operator S x , y is its minimum eigenvalue λ min ( S x , y ) . A large minimum eigen-value means that Alice has managed to force the outcome 1 to appear, regardless of what20tate Bob plays, whereas a small minimum eigenvalue means that Bob has at least onechoice of a state that causes the outcome 1 to appear with small probability. Stated inmore precise terms, Bob’s optimal strategy in the case that Alice plays ρ = | y (cid:105)(cid:104) y | is toplay any state σ ∈ D ( B ) whose image is contained in the eigenspace of S x , y correspond-ing to the minimum eigenvalue λ min ( S x , y ) , which leads to a win for Alice with probabilityequal to this minimum eigenvalue and a win for Bob with probability 1 − λ min ( S x , y ) .In general, Alice will not play deterministically, but will instead play a distributionof strings p ∈ P ( Σ n ) . In this case, the resulting measurement operator on Bob’s spacebecomes ∑ y ∈ Σ n p ( y ) S x , y . (69)That is to say, the probability that Alice wins when she plays a distribution p ∈ P ( Σ n ) ,and Bob plays optimally against this distribution, is given by the expression λ min (cid:32) ∑ y ∈ Σ n p ( y ) S x , y (cid:33) . (70)Determining whether x is a yes-instance or a no-instance of A is therefore equivalent todiscriminating between the case that there exists a distribution p ∈ P ( Σ n ) for which theminimum eigenvalue (70) is at least 3/4 and the case in which this minimum eigenvalueis at most 1/4 for all choices of p ∈ P ( Σ n ) .The goal of the proof is to show that this decision problem is contained in ∃ · PP. The ∃ operator will represent the existence or non-existence of a distribution p ∈ P ( Σ n ) forwhich the minimum eigenvalue (70) is large, while a PP predicate will allow for an esti-mation of this minimum eigenvalue itself. A challenge that must be overcome in makingthis approach work is that using the ∃ operator in this way requires Alice’s strategy tohave a polynomial-length representation. However, given that a distribution p ∈ P ( Σ n ) may have support that is exponentially large in n , an explicit description of p will gener-ally have exponential size, assuming that the individual probabilities p ( y ) are representedwith a polynomial number of bits of precision.This obstacle may be overcome using the Althöfer–Lipton–Young [Alt94, LY94] tech-nique mentioned in the introduction: in place of a distribution p ∈ P ( Σ n ) , we consideran N -tuple of strings ( y , . . . , y N ) , representing N possible deterministic plays for Alice,for N = N ( | x | ) being a suitable polynomially bounded function of the input length. This N -tuple will represent the distribution q ∈ P ( Σ n ) obtained by selecting j ∈ {
1, . . . , N } uniformly at random and then outputting the string y j . That is, the distribution q ∈ P ( Σ n ) represented by the N -tuple ( y , . . . , y N ) is given by q ( y ) = (cid:12)(cid:12) { j ∈ {
1, . . . , N } : y = y j } (cid:12)(cid:12) N (71)for each y ∈ Σ n . Naturally, most choices of a distribution p ∈ P ( Σ n ) are far away fromany such distribution q . Nevertheless, the existence of a distribution p ∈ P ( Σ n ) for which21 B P x ρ Figure 3: Definition 15 is concerned with the probability that the output of a circuit P x ,measured with respect to the standard basis, is contained in the language B , assumingthe input is ρ .the minimum eigenvalue (70) is large does in fact imply the existence of an N -tuple ( y , . . . , y N ) for which the distribution q ∈ P ( Σ n ) defined by (71) is still a good play forAlice, meaning that the minimum eigenvalue λ min (cid:18) S x , y + · · · + S x , y N N (cid:19) (72)is also large, provided N is sufficiently large. This is precisely the content of Lemma 12.In particular, by choosing N = ( m + ) , where m is the number of qubits of B , wefind that if the minimum eigenvalue (70) is at least 3/4, then with probability at least2/3 (over the random choices of y , . . . , y N ) the minimum eigenvalue (70) is at least 2/3.Of course, this implies the existence of an N -tuple ( y , . . . , y N ) for which the minimumeigenvalue (70) is at least 2/3.Naturally, if x ∈ A no , then the minimum eigenvalue (70) is at most 1/4 for all choicesof p ∈ P ( Σ n ) , and therefore it must be that λ min (cid:18) S x , y + · · · + S x , y N N (cid:19) ≤ <
13 (73)for all N -tuples ( y , . . . , y N ) . This is because the distribution q defined by (71) is simplyone example of a distribution in P ( Σ n ) .The purpose of Lemma 13 is now evident, for it states that there exists a language B ∈ PP such that if the minimum eigenvalue (72) is at least 2/3, then ( x , y · · · y N ) ∈ B ,while if this minimum eigenvalue is at most 1/3, then ( x , y · · · y N ) (cid:54)∈ B . Consequently, if x ∈ A yes , then there exists a string y · · · y N ∈ Σ nN such that ( x , y · · · y N ) ∈ B , while if x ∈ A no , then for every string y · · · y N ∈ Σ nN it is the case that ( x , y · · · y N ) (cid:54)∈ B . It hastherefore been proved that A ∈ ∃ · PP as required.
5. Upper-bound on MQRG(1)
We now turn to the complexity class MQRG ( ) , and prove the containment MQRG ( ) ⊆ P · PP. In order to do this, we will first introduce a QMA-operator that, in some sense,functions in a way that is similar to the ∃ and P operators previously discussed.22 efinition 15. For a given complexity class C , the complexity class QMA · C contains allpromise problems A = ( A yes , A no ) for which there exists a polynomial-time generatedfamily of quantum circuits { P x : x ∈ Σ ∗ } , where each P x takes n = n ( | x | ) input qubitsand outputs k = k ( | x | ) qubits, along with a language B ∈ C , such that the followingimplications hold.1. If x ∈ A yes , then there exists a density operator ρ on n qubits for whichPr (cid:0) P x ( ρ ) ∈ B (cid:1) ≥
23 . (74)2. If x ∈ A no , then for every density operator ρ on n qubits,Pr (cid:0) P x ( ρ ) ∈ B (cid:1) ≤
13 . (75)Here, the notation P x ( ρ ) ∈ B refers to the event that P x is applied to the state ρ , theoutput qubits are measured with respect to the standard basis, and the resulting string iscontained in the language B . Figure 3 illustrates the associated process, with χ B being thecharacteristic function of B on inputs of length k . Theorem 16. If C is nontrivial complexity class of languages that is closed under joins and truth-table reductions, then QMA · C ⊆ P · C .Proof. Let A = ( A yes , A no ) ∈ QMA · C , and let { P x : x ∈ Σ ∗ } be a polynomial-timegenerated family of quantum circuits that, together with a language B ∈ C , establishesthis inclusion according to Definition 15.By Lemma 7 there exists a polynomially bounded function r and GapP functions f and f such that Re (cid:0) (cid:104) u | P x (cid:0) | z (cid:105)(cid:104) w | (cid:1) | v (cid:105) (cid:1) = − r f ( x , z , w , u , v ) ,Im (cid:0) (cid:104) u | P x (cid:0) | z (cid:105)(cid:104) w | (cid:1) | v (cid:105) (cid:1) = − r g ( x , z , w , u , v ) , (76)for all x ∈ Σ ∗ , z , w ∈ Σ n , and u , v ∈ Σ k . Define g ( x , z , w , u ) = (cid:40) f ( x , z , w , u , u ) if u ∈ B u (cid:54)∈ B , g ( x , z , w , u ) = (cid:40) f ( x , z , w , u , u ) if u ∈ B u (cid:54)∈ B , (77)for all x ∈ Σ ∗ , z , w ∈ Σ n , and u ∈ Σ k . By the nontriviality and closure of C under Karpreductions (the full power of closure under joins and truth-table reductions is not requiredfor this step), it is the case that g , g ∈ Gap · C .23ext, define F ( x , z , w ) = ∑ u ∈ Σ k g ( x , z , w , u ) , F ( x , z , w ) = − ∑ u ∈ Σ k g ( x , z , w , u ) , (78)for all x ∈ Σ ∗ and z , w ∈ Σ n . By Lemma 4 we have that F , F ∈ Gap · C . We observe thatRe (cid:0) (cid:104) z | R x | w (cid:105) (cid:1) = − r F ( x , z , w ) ,Im (cid:0) (cid:104) z | R x | w (cid:105) (cid:1) = − r F ( x , z , w ) (79)for all x ∈ Σ ∗ and z , w ∈ Σ n , where R x = ∑ u ∈ Σ k ∩ B P ∗ x (cid:0) | u (cid:105)(cid:104) u | (cid:1) . (80)Now let us consider the cases x ∈ A yes and x ∈ A no . If x ∈ A yes then λ max ( R x ) ≥ x ∈ A no then λ max ( R x ) ≤ R x is a positive semidefiniteoperator on a 2 n dimensional space, we have that λ max ( R x ) n + = λ max ( R n + x ) ≤ Tr ( R n + x ) ≤ n λ max ( R n + x ) = n λ max ( R x ) n + , (81)similar to equation (53) in the proof of Lemma 13. By Lemma 6 it follows that there existsa Gap · C function G possessing the following properties.1. If x ∈ A yes then G ( x ) = ( n + ) r tr ( R n + x ) ≥ ( n + ) r + n + n + (82)2. If x ∈ A no then G ( x ) = ( n + ) r tr ( R n + x ) ≤ ( n + ) r + n n + . (83)The Gap · C function H ( x ) = n + G ( x ) − ( n + ) r + n (84)therefore satisfies H ( x ) > x ∈ A yes and H ( x ) ≤ x ∈ A no . By Proposition 3it follows that A ∈ P · C .Next, we prove that MQRG ( ) is contained in QMA · PP. Combining this fact with theprevious theorem will establish the main result as an immediate corollary.
Theorem 17.
MQRG ( ) ⊆ QMA · PP .Proof. Consider any promise problem A = ( A yes , A no ) in MQRG(1), and fix a referee thatestablishes the inclusion A ∈ MQRG(1) . Let { P x : x ∈ Σ ∗ } and { Q x : x ∈ Σ ∗ } be acollection of circuits that describe this referee, in accordance with Definition 11. As in theproof of Theorem 14, define an operator S x , y = (cid:0) (cid:104) y | ⊗ B (cid:1) Q ∗ x ( | (cid:105)(cid:104) | ) (cid:0) | y (cid:105) ⊗ B (cid:1) (85)24or each x ∈ Σ ∗ and y ∈ Σ k . If x ∈ A yes , there must exists a state ρ ∈ D ( A ) such that λ min (cid:32) ∑ y ∈ Σ k (cid:104) y | P x ( ρ ) | y (cid:105) S x , y (cid:33) ≥
34 , (86)while if x ∈ A no , it is the case that λ min (cid:32) ∑ y ∈ Σ k (cid:104) y | P x ( ρ ) | y (cid:105) S x , y (cid:33) ≤
14 (87)for every ρ ∈ D ( A ) .Now define a function N = ( m + ) and observe that N is polynomially boundedin | x | . By Lemma 13, there exists a language B ∈ PP for which these implications hold forall x ∈ Σ ∗ and y , . . . , y N ∈ Σ k : λ min (cid:18) S x , y + · · · + S x , y N N (cid:19) ≥ ⇒ ( x , y · · · y N ) ∈ B , (88) λ min (cid:18) S x , y + · · · + S x , y N N (cid:19) ≤ ⇒ ( x , y · · · y N ) (cid:54)∈ B . (89)Finally, for each input x , define a circuit K x that takes as input N registers ( A , . . . , A N ) ,each consisting of n qubits, and outputs N + ( X , Y , . . . , Y N ) . The register X isinitialized to the state | x (cid:105)(cid:104) x | , so that it simply echoes the input string x , and each register Y j is obtained by independently applying the circuit P x to A j . Alternatively, one couldwrite K x = | x (cid:105)(cid:104) x | ⊗ P ⊗ Nx , (90)with the understanding that we are identifying the state | x (cid:105)(cid:104) x | with the channel thatinputs nothing and outputs the state | x (cid:105)(cid:104) x | .To prove that the promise problem A is contained in QMA · PP, it suffices to provetwo things:
Completeness.
If it is the case that x ∈ A yes , then there must exist a state ξ ∈ D ( A ⊗ N ) suchthat Pr ( K x ( ξ ) ∈ B ) ≥
23 . (91)
Soundness.
If it is the case that x ∈ A no , then for every state ξ ∈ D ( A ⊗ N ) it must be thatPr ( K x ( ξ ) ∈ B ) ≤
13 . (92)The proof of completeness follows a similar argument to the proof of Theorem 14. Let ρ ∈ D ( A ) be any state for which (86) is satisfied, and let ξ = ρ ⊗ N . It is evident that theoutput of K x ( ξ ) is given by ( x , y · · · y N ) , for y , . . . , y N ∈ Σ k sampled independently fromthe distribution p ( y ) = (cid:104) y | P x ( ρ ) | y (cid:105) . (93)25t follows by Lemma 12 that Pr ( K x ( ξ ) ∈ B ) ≥
23 . (94)For the proof of soundness, the possibility that the state ξ ∈ D ( A ⊗ N ) does not takeproduct form must be considered. Our aim is to prove that if y , . . . , y N are randomlyselected according to the distribution that assigns the probability (cid:10) y · · · y N (cid:12)(cid:12) P ⊗ Nx ( ξ ) (cid:12)(cid:12) y · · · y N (cid:11) (95)to each tuple ( y , . . . , y N ) , thenPr (cid:32) λ min (cid:32) S x , y + · · · + S x , y N N (cid:33) ≤ (cid:33) ≥
23 , (96)for this implies that Pr ( K x ( ξ ) ∈ B ) ≤ σ ∈ D ( B ) for which ∑ y ∈ Σ k (cid:104) y | P x ( ρ ) | y (cid:105) Tr (cid:0) S x , y σ (cid:1) ≤
14 (97)for all ρ ∈ D ( A ) , which is possible by Sion’s min-max theorem under the assumption (87),and define random variables Z , . . . , Z N as Z j = Tr (cid:0) S x , y j σ (cid:1) (98)for every j ∈ {
1, . . . , N } , assuming that y , . . . , y N are chosen at random as above. It suf-fices to prove that Pr (cid:32) Z + · · · + Z N N ≤ (cid:33) ≥
23 , (99)as we have λ min ( H ) ≤ Tr ( H σ ) for all Hermitian operators H .The complication we face at this point is that the random variables Z , . . . , Z N are notnecessarily independent (because ξ does not necessarily have product form), so the moststandard form of Hoeffding’s inequality will not suffice to establish the required bound(99). However, we observe that Z , . . . , Z N are discrete random variables that take valuesin the interval [
0, 1 ] and satisfy the inequalityE ( Z j | Z = α , . . . , Z j − = α j − ) ≤
14 (100)for all j ∈ {
2, . . . , N } and α , . . . , α j − ∈ [
0, 1 ] for which Pr ( Z = α , . . . , Z j − = α j − ) is nonzero. This is evident from the inequality (97), for it must hold when ρ is equal tothe reduced state of register A j , conditioned on any choice of y , . . . , y j − (and thereforeon any choice of values Z = α , . . . , Z j − = α j − ) that appear with nonzero probability.26hile the standard statement of Hoeffding’s inequality does not suffice for our needs, thestandard proof of Hoeffding’s inequality does establish thatPr (cid:32) Z + · · · + Z N N ≥ (cid:33) = Pr (cid:32) Z + · · · + Z N N ≥ + (cid:33) ≤ exp (cid:18) − N (cid:19) <
13 , (101)as explained in an appendix at the end of the paper. Having obtained this bound, theproof is complete.
Corollary 18.
MQRG ( ) ⊆ P · PP .
6. Conclusion
We have proved containments on two restricted versions of QRG(1), which we have calledCQRG(1) and MQRG(1). An obvious challenge is to prove a stronger containment onthe class QRG(1) than PSPACE. Observing that the containments we prove establish thatCQRG(1) and MQRG(1) are contained in the counting hierarchy, we wonder whetherQRG(1) is also contained in the counting hierarchy.
A. Hoeffding’s inequality for dependent random variableswith bounded conditional expectation
In the proof of Theorem 17 we used a slight variant of Hoeffding’s inequality, where theassumption of independence is replaced by a bound on conditional expectation. We ex-pect that a bound along these lines has been observed before, but we have not found asuitable reference. (A similar bound is proved in [BCF +
95] for Bernoulli random vari-ables, but we require the bound to hold more generally for discrete random variables.)It is, however, straightforward to adapt the most typical proof of Hoeffding’s inequal-ity to obtain this bound, as we now explain. We begin with Hoeffding’s lemma, which isthe essential ingredient in the proof, and which we state without proof. (A proof may befound in [BW16], among many other references.)
Lemma 19 (Hoeffding’s lemma) . Let X be a random variable taking values in [ α , β ] , for realnumbers α < β , and assume E ( X ) ≤ . For every λ > it is the case that E (cid:0) exp ( λ X ) (cid:1) ≤ exp (cid:18) λ ( β − α ) (cid:19) . (102) Remark 20.
The more typical assumption for this lemma is that E ( X ) =
0, but (as is notsurprising) it is true assuming instead that E ( X ) ≤
0. This follows immediately from theobservation that if E ( X ) ≤
0, thenE ( exp ( λ X )) ≤ E ( exp ( λ ( X − E ( X )))) . (103)27he next lemma provides the inequality in the proof of Hoeffding’s inequality thatwould ordinarily follow from the assumption of independence. For simplicity we provethis lemma for discrete random variables, which suffices for our needs. Lemma 21.
Let X and Y be discrete random variables taking values in [ α , β ] for real numbers α < β , and assume that E ( Y | X ) ≤ . For every λ > it is the case that E ( exp ( λ ( X + Y )) ≤ exp (cid:18) λ ( β − α ) (cid:19) E ( exp ( λ X )) . (104) Proof.
We may writeE ( exp ( λ ( X + Y )) = ∑ x exp ( λ x ) E ( exp ( λ Y ) | X = x ) Pr ( X = x ) , (105)where the sum ranges over all possible values of X . By the assumption E ( Y | X ) ≤ ∑ x exp ( λ x ) E ( exp ( λ Y ) | X = x ) Pr ( X = x ) ≤ exp (cid:18) λ ( β − α ) (cid:19) ∑ x exp ( λ x ) Pr ( X = x ) = exp (cid:18) λ ( β − α ) (cid:19) E ( exp ( λ X )) , (106)as required.Finally, we state and prove the variant of Hoeffding’s inequality we have used (againfor discrete random variables). Theorem 22.
Let X , . . . , X n be discrete random variables taking values in [
0, 1 ] , let γ ∈ [
0, 1 ] ,and assume that E ( X k | X , . . . , X k − ) ≤ γ (107) for all k ∈ {
1, . . . , n } . For all ε > it is the case that Pr (cid:0) X + · · · + X n ≥ ( γ + ε ) n (cid:1) ≤ exp ( − n ε ) . (108) Proof.
For every λ > (cid:0) X + · · · + X n ≥ ( γ + ε ) n (cid:1) = Pr (cid:0) exp (cid:0) λ ( X + · · · + X n − γ n ) (cid:1) ≥ exp ( λε n ) (cid:1) ≤ E (cid:0) exp (cid:0) λ ( X + · · · + X n − γ n ) (cid:1)(cid:1) exp ( λε n ) (109)by Markov’s inequality. Applying Lemma 21 iteratively yieldsE (cid:0) exp (cid:0) λ ( X + · · · + X n − γ n ) (cid:1)(cid:1) ≤ exp (cid:18) n λ (cid:19) . (110)Choosing λ = ε yields the claimed bound.28 eferences [AKN98] D. Aharonov, A. Kitaev, and N. Nisan. Quantum circuits with mixed states. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing , pages20–30, 1998.[Alt94] I. Althöfer. On sparse approximations to randomized strategies and convexcombinations.
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