Quantum interactive proofs with short messages

This paper considers three variants of quantum interactive proof systems in which short (meaning logarithmic-length) messages are exchanged between the prover and verifier. The first variant is one in which the verifier sends a short message to the prover, and the prover responds with an ordinary, or polynomial-length, message; the second variant is one in which any number of messages can be exchanged, but where the combined length of all the messages is logarithmic; and the third variant is one in which the verifier sends polynomially many random bits to the prover, who responds with a short quantum message. We prove that in all of these cases the short messages can be eliminated without changing the power of the model, so the first variant has the expressive power of QMA and the second and third variants have the expressive power of BQP. These facts are proved through the use of quantum state tomography, along with the finite quantum de Finetti theorem for the first variant.


Introduction
The interactive proof system model extends the notion of efficient proof verification to an interactive setting, where a computationally unrestricted prover tries to convince a computationally bounded verifier that an input string satisfies a particular fixed property. They have been studied extensively in computational complexity theory since their introduction roughly 25 years ago [GMR85,GMR89,Bab85,BM88], and as a result much is known about them. (See [AB09] and [Gol08], for instance, for further discussions of classical interactive proof systems.) Quantum interactive proof systems are a natural quantum computational extension of the interactive proof system model, where the prover and verifier can perform quantum computations and exchange quantum information. The expressive power of quantum interactive proofs is no different from classical interactive proofs: it holds that QIP = PSPACE = IP, and therefore any problem having a quantum interactive proof system also has a classical one [JJUW09,LFKN92,Sha92]. However, quantum interactive proof systems may be significantly more efficient than classical interactive proofs in terms of the number of messages they require, as every problem in PSPACE has a quantum interactive proof system requiring just three messages to be exchanged between a prover and verifier [KW00]. This is not possible classically unless AM = PSPACE, and this equality implies the collapse of the polynomial-time hierarchy [BM88,GS89].
In this paper we consider quantum interactive proof systems in which some of the messages are short, by which we mean that the messages consist of a number of qubits that is logarithmic in the input length. Three particular variants of quantum interactive proofs with short messages are considered. The first variant is one in which the verifier sends a short message to the prover, and the prover responds with an ordinary, or polynomial-length, message. We prove that this model has the expressive power of QMA. The second variant is one in which any number of messages can be exchanged between the prover and verifier, but where the combined length of all the messages is logarithmic. We prove that this model has the expressive power of BQP. The third variant is one in which the verifier sends polynomially many random bits to the prover, who responds with a short quantum message. We prove that this model also has the expressive power of BQP. Thus, in each of these three cases, logarithmic-length messages are effectively worthless and can be removed without changing the power of the model.
One possible application of our work is to the design of new quantum algorithms or QMA verification procedures. Although we do not yet have interesting examples, we believe it is possible that an intuition about quantum interactive proof systems with short messages may lead to new problems being shown to be in BQP or QMA, based on characterizations of the sort we prove.
Observe that all of these three results are immediate in the classical case. For example, one can enumerate all logarithmic-length interactions between a verifier and prover in polynomialtime, so our second model, assuming that the verifier is classical, has the expressive power of P (or BPP in the presence of randomness). This argument, however, does not work in the quantum case. To explain the difference let us consider the following simplification of this model. Assume that instead of an arbitrary number of messages of logarithmic total length, there is only one logarithmic-size message allowed which is sent by the prover. This model is denoted by QMA log , and was known to be equal to BQP [MW05]. Here we present another proof for this fact to illustrate the main ideas of the paper. Upon receiving a logarithmic-size message from the prover, the verifier applies a binary measurement {P acc , P rej } to decide whether to accept or reject. Thus the acceptance probability is at most the maximum eigenvalue of P acc . Although P acc acts on a logarithmic number of qubits, it is given by a polynomial-size circuit, so one cannot directly compute the matrix representation of P acc in polynomial-time. Nevertheless, using quantum process tomography we can perform the measurement {P acc , P rej } on polynomially many known states, and somehow by taking the average of their outcomes compute an approximation of P acc . Since the matrix representation of P acc has only polynomially many entries, this approximation can be arbitrarily tight. The next step is to simply find the maximum eigenvalue of this approximation.
In this paper instead of applying quantum process tomography on a measurement, we perform quantum state tomography on the normalized Choi-Jamiołkowski representation of the quantum channel corresponding to the measurement. These two approaches are equivalent, but the second one unifies the arguments in different sections.
Besides quantum state tomography and Choi-Jamiołkowski representation of quantum channels, finite quantum de Finetti theorem is another important tool in this work. Suppose that we are given some copies of a state and we want to verify that it is closed to some given state. Using quantum state tomography on these copies we can find an approximation of the unknown state and solve the problem. Assume now that we are not guaranteed that these copies are indeed copies of the same state; there can even be entanglement among different copies. To overcome these difficulties we use finite quantum de Finetti theorem to reduce the problem to the first case.
The remainder of this paper has the following organization. Section 2 discusses some of the background information needed for the rest of the paper, including background on the Choi-Jamiołkowski representation of quantum channels, quantum state tomography, finite quantum de Finetti theorem, and quantum interactive proof systems. Sections 3, 4, and 5 then discuss the three variants of quantum interactive proof systems with short messages described above.

Background
We assume the reader is familiar with quantum information and computation, including the basic quantum complexity classes BQP and QMA, simple properties of mixed states, measurements, channels, and so on [KSV02,NC00]. The purpose of the present section is to highlight background knowledge on three topics, represented by the three subsections below, that are particularly relevant to this paper. These topics are: the Choi-Jamiołkowski representation of quantum channels, quantum state tomography, and quantum interactive proof systems.
Before discussing these three topics, it is appropriate to mention a few simple points of notation and terminology. Throughout this paper we let Σ = {0, 1} denote the binary alphabet, and for each k ∈ N we write C(Σ k ) to denote the finite-dimensional Hilbert space whose standard basis vectors are indexed by Σ k (i.e., the Hilbert space associated with a k-qubit quantum register). The Dirac notation is used to describe vectors in spaces of this sort.
For a given space Q = C(Σ k ), we write L (Q) to denote the space of all linear mappings from Q to itself, which is associated with the space of all complex matrices with rows and columns indexed by Σ k in the usual way. The subsets of this space representing the positive semidefinite operators and density operators on Q are denoted Pos (Q) and D (Q), respectively. A standard inner product on L (Q) is defined as X, Y = Tr(X * Y) for all X, Y ∈ L (Q) (where X * denotes the adjoint, or conjugate-transpose, of X). The trace norm of an operator X ∈ L (Q) is defined as and the spectral (or operator) norm of X is denoted X .

Quantum channels and the Choi-Jamiołkowski representation
A quantum channel from a k-qubit space Q = C(Σ k ) to an l-qubit space R = C(Σ l ) is a completely positive and trace-preserving linear mapping of the form Φ : , for every Hilbert space S, is positive, meaning that it sends positive semidefinite operators to positive semidefinite ones. Trance-preserving means that Tr(Φ(ρ)) = Tr(ρ).) We will write C (Q, R) to denote the set of all such quantum channels. For any quantum channel Φ ∈ C (Q, R) one defines the (normalized) Choi-Jamiołkowski representation [Jam72, Cho75] of Φ as In other words, this is the l + k qubit state that results from applying Φ to one-half of k pairs of qubits in the |φ + = (|00 + |11 )/ √ 2 state.
The action of the mapping Φ can be recovered from its normalized Choi-Jamiołkowski representation in the following way that makes use of post-selection. Suppose that Q and Q 0 are k-qubit registers and R is an l-qubit register, that the pair (R, Q 0 ) is initialized to the state ρ as defined by Φ in (1), and that Q is in an arbitrary quantum state (and is possibly entangled with additional registers not including Q 0 and R). Consider the following procedure: 1. Measure each qubit of Q together with its corresponding qubit in Q 0 with respect to the Bell basis.
2. If every one of these k measurements results in an outcome corresponding to the Bell state |φ + , then output "success," else output "failure." This procedure gives the outcome "success" with probability 4 −k , and conditioned on success the register R is precisely as it would be had it resulted from the channel Φ being applied to Q. (The registers Q and Q 0 can safely be discarded if the procedure succeeds.) To see this, assume first that the joint state of (R, Q 0 , Q) is ρ ⊗ ξ before the measurement takes place. Then the (unnormalized) state of R after the measurements are performed, assuming the end result is "success," is The probability of success is therefore 4 −k , and conditioned on this outcome the process implements the channel Φ. In our applications k is logarithmic in the size of the problem, so Φ is implemented with an inverse polynomial probability which is enough for us. The fact that this process implements the channel Φ exactly for all density operators ξ implies that it also operates correctly in the case that Q is entangled with additional registers.

Quantum state tomography
Quantum state tomography is the process by which an approximate description of an unknown quantum state is obtained by measurements on many independent copies of the unknown state.
To be more precise, let Q = C(Σ k ) denote the space corresponding to a k-qubit register, and suppose that X 1 , . . . , X N are k-qubit quantum registers independently prepared in an unknown k-qubit state ρ ∈ D (Q). The purpose of quantum state tomography is to obtain an explicit description of a k-qubit state that closely approximates ρ. One way to perform quantum state tomography is through the use of an information-complete measurement. A measurement {P a : a ∈ Γ} on k-qubit registers is information-complete if and only if the set {P a : a ∈ Γ} spans the entire 4 k -dimensional space L (Q). When such a measurement is performed on a k-qubit state ρ, each measurement outcome is obtained with probability Based on the assumption that {P a : a ∈ Γ} is information-complete, this vector p of probabilities uniquely determines the state ρ. A close approximation of p, which may be obtained by sufficiently many independent measurements, leads to an approximate description of ρ.
The accuracy of an approximation based on the process just described naturally depends on the choice of an information-complete measurement as well as the specific notion of approximation that is considered. Our interest will be on the trace distance ρ − σ 1 between the approximation σ and the true state ρ. To describe the "quality" of an information-complete measurement, it is appropriate to describe the specific process that is used to reconstruct ρ from the vector of probabilities p.
For any spanning set {P a : a ∈ Γ} of L (Q), there exists a set {M a : a ∈ Γ} ⊆ L (Q) that satisfies ∑ a∈Γ M a P a , X = X for every X ∈ L (Q). (One may find such a set {M a : a ∈ Γ} by solving a system of linear equations.) The set {M a : a ∈ Γ} is uniquely determined when {P a : a ∈ Γ} has exactly 4 k elements (i.e., is a basis), and hereafter we will restrict our attention to this case. Notice that if ρ is a density matrix, the coefficients p(a) = P a , ρ form a probability distribution. If q is a probability vector that represents an approximation to p, it holds that It is therefore desirable that the maximum trace norm over the set {M a : a ∈ Γ} determined by the measurement {P a : a ∈ Γ} is as small as possible.
There is one additional consideration that is sometimes relevant, which is that the approximation ∑ a∈Γ q(a)M a may fail to be positive semidefinite, and therefore fail to represent a valid quantum state. In this situation one can find a quantum state near to the approximation by renormalizing the positive part of the approximation. For the applications of tomography in this paper, however, this issue may safely be disregarded, as non-positive approximations of density operators will still provide valid approximations to the quantities we are interested in.
An example of an information-complete measurement on a single qubit is given by the following matrices: This is not an optimal information-complete measurement, but it has the advantage of being simple to describe and can be implemented exactly by a quantum circuit composed of Hadamard, controlled-not, and π/8-phase gates, and measurement in the standard basis. The corresponding set {M 0 , M 1 , M 2 , M 3 } described above is given by It holds that M a 1 = √ 10 < 4 for a ∈ Γ = {0, 1, 2, 3}. An information-complete measurement for k qubits may be obtained by taking tensor products of the above matrices. More specifically, for each x ∈ Γ k , let us define 2 k × 2 k matrices P x and M x as Then {P x : x ∈ Γ k } is an information-complete measurement, and its corresponding set is given by {M x : x ∈ Γ k }. By the multiplicativity of the trace norm, it holds that M x 1 = 10 k/2 < 4 k for every k. Now, let us suppose that ρ is a quantum state on k qubits, and tomography (using the measurements just described) is performed on N copies of ρ. More precisely, the measurement {P x } is performed independently on each of the N copies of ρ, a probability distribution q : Γ k → [0, 1] is taken to be the frequency distribution of the outcomes, and an approximation to ρ is computed. We require a bound on the accuracy of this approximation. Of course, nothing can be said in the worst case, as any sequence of measurement outcomes could occur with very small probability in general.
Proof. For any δ > 0, and any fixed choice of x ∈ Γ k , it follows from Hoeffding's inequality that By the union bound it follows that Setting δ = ε/16 k and using the inequality e −α < 1/α for all α > 0, we have The notion of quantum process tomography has also been considered, where a quantum measurement or channel is approximated through many independent evaluations of an appropriate sort (see for example [NC00]). In this paper, however, it is not necessary to consider this sort of tomography as being any different from state tomography. Specifically, we will approximate channels (and measurements, modeled as channels) by evaluating them on maximally entangled states, followed by ordinary quantum state tomography on the normalized Choi-Jamiołkowski representations that result.

Finite quantum de Finetti theorem
Suppose that Q 1 , . . . , Q N are all k-qubit registers. A state on (Q 1 , . . . , Q N ) is called symmetric if it is invariant under any permutation of its registers. For instance, any product state of the form ρ ⊗N is symmetric. Any convex combination of such states is symmetric as well. Note, however, that there are symmetric states that cannot be written as a convex combination of symmetric product states as above; as an example consider the following state Nevertheless, by tracing out any subsystem of |ψ the resulting reduce density matrix is in the convex hall of symmetric product states. The following theorem generalizes this observation.

Quantum interactive proofs
Quantum interactive proof systems are a natural quantum analogue of ordinary, classical interactive proof systems, where the prover and verifier may process and exchange quantum information. We will only consider quantum interactive proof systems having an even number of messages in this paper, so for simplicity we will restrict our discussion to this case. For t being a function of the form t : N → N, we define a t-round (or (2t)-message) quantum verifier V to be a collection of quantum circuits which can be generated in polynomial-time given x and j. We will generally write t rather than t(| x|) hereafter in this paper, keeping in mind that t might vary with the input length. We assume that the verifier's circuits are composed of standard unitary quantum gates (controlled-not, Hadamard, and π/8-phase gates, let us say), as well as ancillary and erasure gates. Included in the description of these circuits is a specification of which input and output qubits are to be considered private memory qubits and which are considered message qubits. The message qubits refer to qubits that are sent to or received from a prover (to be described shortly). The following properties are required of the circuits describing a verifier: 1. For each x, the circuit V x,0 takes no input qubits, and the circuit V x,t produces a single output qubit (called the acceptance qubit).
2. There exist functions v 1 , v 2 , . . . such that V x,j−1 outputs v j (|x|) private memory qubits and V x,j inputs v j (| x|) private memory qubits for 1 ≤ j ≤ t.
3. There exist functions q 1 , q 2 , . . . and r 1 , r 2 , . . . that specify the number of message qubits the verifier sends to or receives from the prover on each round, for a given input length. More precisely, each circuit V x,j−1 outputs q j (| x|) message qubits and each circuit V x,j inputs r j (|x|) message qubits, for 1 ≤ j ≤ t. Figure 1: An illustration of an interaction between a prover and verifier in a quantum interactive proof system. In the picture it is assumed that t = 3. The labels v j , p j , q j and r j on the arrows refer to the number of qubits represented by each arrow.
Similar to the function t, we will often omit the argument |x| from the functions v j , q j , and r j for the sake of readability. When it is convenient, we will refer to the message qubits sent from the verifier to the prover as question qubits and qubits sent from the prover to the verifier as response qubits.
A t-round (or (2t)-message) prover is defined in a similar way to a t-round verifier, but no computational restrictions are made. Specifically, a t-round prover is a collection of quantum channels Again, the input and output qubits of these channels are specified as private memory qubits or message qubits. When a particular prover P is considered to interact with a given verifier V, one naturally assumes that they agree on the number of messages and the number of qubits sent in each message, as suggested by Figure 1. But given the verifier there is no restriction on p j , the number of private memory qubits used by the prover at the j-th round. Although p j could in principle be unbounded, it is not difficult to show that for any choice of verifier and prover there is another prover that simulates the same interaction and uses at most a polynomial number of private memory qubits (see [GW07]). Now, on a given input string x, the prover P and verifier V have an interaction by composing their circuits/channels as described in Figure 1. The maximum acceptance probability for a given verifier V on an input x refers to the maximum probability for the circuit V x,t to output 1, assuming it is measured in the standard basis, over all choices of a compatible prover P. It is always the case that a maximal probability is achieved by some prover.
Classes of promise problems may be defined by quantum interactive proof systems in a variety of ways. We will delay the definitions of the classes we consider to the individual sections in which they are discussed.

Two-message quantum interactive proofs with short questions
The first specific variant of quantum interactive proof systems we consider are those in which just a single round of communication takes place, with the first message being short (at most logarithmic-length) and the second message being normal (at most polynomial-length). In particular, let us say that a 1-round verifier V is a [log, poly] quantum verifier if the number q = q 1 of question qubits it sends during the first and only round of communication satisfies q(n) = O(log n). For functions of the form a, b : N → [0, 1] we define QIP([log, poly], a, b) to be the class of all promise problems B = (B yes , B no ) for which there exists a [log, poly] quantum verifier V with completeness and soundness probability bounds a and b, respectively. In other words, V satisfies the following properties: 1. For every string x ∈ B yes , there exists a prover P compatible with V that causes V to accept x with probability at least a(| x|).
2. For every string x ∈ B no , and every prover P compatible with V, it holds that P causes V to accept x with probability at most b(| x|).
For a wide range of choices of a and b, these classes coincide with QMA as the following theorem states. , and let V be a [log, poly] verifier that witnesses this fact. We write q (as above) to denote the number of question qubits the verifier V sends, and write r to denote the number of response qubits V receives. As V is a [log, poly] verifier it holds that q(n) = O(log n). For a fixed input x, we will write Q = C(Σ q ) to denote the question space and R = C(Σ r ) to denote the response space for V, corresponding to the question and response qubits in the obvious way.
Our goal is to prove that B ∈ QMA, and to do this we will define a verification procedure (to be referred to as Arthur) that demonstrates this fact. Suppose P is a prover that interacts with V. For a fixed input string x, the action of P may be identified with a quantum channel Φ ∈ C (Q, R), and any such channel defines a quantum state ρ ∈ D (R ⊗ Q) according to its normalized Choi-Jamiołkowski representation (1). We will define Arthur so that he expects to receive many independent copies of this state. He will check its validity using quantum state tomography, and will use the state to apply the mapping Φ himself through post-selection.
More specifically, we define Arthur so that he performs the following actions: 1. Input N + m registers (R 1 , Q 1 ), . . . , (R N+m , Q N+m ), where N and m are polynomials in the input length n to be specified below.
3. Perform quantum state tomography on the registers (Q 2 , . . . , Q N+1 ), and reject if the resulting approximation is not within trace-distance δ/2 of the completely mixed state 1/2 q , for δ to be specified below.
4. Simulate the original protocol (P, V) by post-selection using the register pair (R 1 , Q 1 ). Reject if the post-selection fails, and otherwise accept or reject as the outcome of the proof system dictates.
To specify N, m and δ, we first set for p being the polynomial whose reciprocal separates the completeness and soundness probability bounds a and b. Now set Given that q is logarithmic, it holds that N, m, 1/ε and 1/δ are polynomially bounded. Suppose first that x ∈ B yes , which implies that there exists a prover P that causes V to accept x with probability at least a. Let Φ denote the quantum channel that describes the behavior of P, and let ρ be the normalized Choi-Jamiołkowski representation of Φ as described in (1). Then for each of the register pairs (R j , Q j ) being prepared independently in the state ρ, it holds that Arthur rejects in step 3 with probability at most δ/2 (Lemma 1), and accepts in step 4 with probability at least a/4 q (conditioned on not having rejected in step 3). Arthur therefore accepts with probability at least Now let us suppose that x ∈ B no . We first consider the situation in which the state of the registers (Q 1 , . . . , Q N+1 ) at the beginning of step 3 has the form for some density operator ξ ∈ D (Q). There are two cases to consider: one is that ξ − 1/2 q 1 < δ and the other is that ξ − 1/2 q 1 ≥ δ. If it is the case that ξ − 1/2 q 1 < δ, then by the Fuchsvan de Graaf inequalities [FvdG99] there must exist a state ρ ∈ D (R ⊗ Q) satisfying Tr R (ρ) = 1/2 q that is within trace distance ε of the state of (R 1 , Q 1 ). To be more precise, consider a fixed purification |ψ of the state of (R 1 , Q 1 ) with an auxiliary register E. Since ξ = Tr R 1 E 1 (|ψ ψ|) has a high fidelity with 1/2 q , and due to the characterization of fidelity in terms of purifications, there exists a purification of 1/2 q over (R, Q, E) that has a large overlap with |ψ . Then ρ can be chosen as the reduce density matrix of this pure state over (R, Q). Now given that x ∈ B no , the state ρ would cause acceptance in step 4 with probability at most b/4 q , and therefore acceptance may occur in the case at hand with probability at most b/4 q + ε. If, on the other hand, it holds that ξ − 1/2 q 1 ≥ δ, then rejection must occur in step 3 with probability at least 1 − δ/2, so Arthur accepts with probability at most δ/2 (which of course is smaller than b/4 q + ε). Thus, in both cases, acceptance occurs with probability at most b/4 q + ε. It follows that if the registers (Q 1 , . . . , Q N+1 ) are, at the beginning of step 3, in any state of the form (i.e., a convex combination of states of the form just discussed), acceptance may occur with probability at most b/4 q + ε. Finally, by the finite quantum de Finetti theorem (Theorem 2) it holds that the state of (Q 1 , . . . , Q N+1 ) after step 2, is within trace-distance ε of a state of the form (2), and therefore the probability of acceptance is at most b/4 q + 2ε in the general case.
Given that a/4 q − ε and b/4 q + 2ε are efficiently computable and separated by the reciprocal of a polynomial, it holds that B is in QMA as claimed.
Figure 2: Illustration of a quantum interactive proof in which the messages are single bits.

Quantum interactive proofs with only short messages
Next we consider quantum interactive proof systems restricted so that the total number of qubits exchanged by the prover and verifier is logarithmic. We prove that any problem having such a quantum interactive proof system is contained in BQP. This fact represents a significant generalization of the equality QMA log = BQP proved in [MW05]. Like the result of the previous section, our proof of this fact is based on quantum state tomography. In addition we will make use of the quantum games framework of [GW07]. It is clear that any quantum interactive proof system allowing at most a logarithmic number of qubits to be exchanged can be simulated by one in which a logarithmic number of single qubit messages are permitted, because any number of these messages could consist of meaningless "dummy" qubits that are interspersed with the qubits sent by the other party. To be more precise, let t(n) = O(log n) and consider a t-round quantum interactive proof system in which each message consists of a single qubit (i.e., q 1 = r 1 = · · · = q t = r t = 1). We will write QIP log (a, b) to denote the class of problems having quantum interactive proof systems of this sort having completeness and soundness probability bounds a and b, respectively. As the following theorem states, this model offers no computational advantage over BQP.
Proof. It is clear that BQP ⊆ QIP log (a, b), and so it remains to prove the reverse containment. To this end let B = (B yes , B no ) be a promise problem in QIP log (a, b), and let V be a verifier that witnesses this fact. As above, let t(n) = O(log n) denote the number of rounds of communication this verifier exchanges with any compatible prover. For a fixed input string x, we will write Q 1 , . . . , Q t to denote copies of the Hilbert spaces C(Σ) associated with the t single-qubit messages that V sends to a given prover P, and we will write R 1 , . . . , R t to denote copies of the same space C(Σ) corresponding to the response qubits of P.
The action of V, on a given input string x, is determined by t + 1 quantum circuits V x,0 , . . . , V x,t as defined in Section 2. Figure 2 illustrates an interaction between V and a prover P for the case that t = 4. Now consider the channel Φ obtained from the circuits V x,0 , . . . , V x,t by setting all of the response qubits the verifier receives from the prover as input qubits and setting all of the question qubits sent by the verifier to the prover as output qubits. More precisely, Φ maps states on the space R 1 ⊗ · · · ⊗ R t to states on the space A ⊗ Q 1 ⊗ · · · ⊗ Q t , where A denotes the singlequbit space associated with the acceptance qubit. Figure 3 illustrates this channel for the protocol The state ρ is obviously efficiently preparable given a description of V. By independently preparing N = 2 10(2t+1) /ε 3 copies of ρ, for ε > 0 to be specified later, and performing quantum state tomography, one obtains a Hermitian operator H on A ⊗ R 1 ⊗ · · · ⊗ R t ⊗ Q 1 ⊗ · · · ⊗ Q t that satisfies H − ρ 1 < ε with probability at least 1 − ε. Let us also define to denote the projection of these operators on the subspace in which the qubit A is |1 (corresponding to accept).
Using the terminology of [GW07], ρ 1 is a co-strategy which describes the verifier's action, and the prover optimizes the acceptance probability corresponding to ρ 1 over all strategies. (Strategies are defined similar to co-strategies as above but with respcet to the prover's action.) Given any strategy X of the prover, the acceptance probability is proportional to the inner product of ρ 1 and X. More precisely, the maximum acceptance probability is equal to maximize: 2 t ρ 1 , X subject to: X ∈ S t where S t ⊂ Pos (R 1 ⊗ · · · ⊗ R t ⊗ Q 1 ⊗ · · · ⊗ Q t ) is the space of all strategies. It is shown in [GW07] that S t is characterizes as S 0 = 1 and for j ≥ 1. This characterization of S t turns the above optimization problem to a semidefinite program. So we just need to replace ρ 1 with its approximation H 1 .
It is clear that Tr(X) = 2 t for every X ∈ S t , and therefore for every X ∈ S t . By taking ε = 1 4 t+1 p for instance, one may therefore distinguish the cases x ∈ B yes and x ∈ B no with probability 1 − ε, by solving the semidefinite program described above. (Semidefinite programs can be efficiently solved up to an inverse polynomial accuracy.) We note that precisely the same argument allows one to conclude that quantum refereed games, as defined in [GW07], allowing for at most a logarithmic number of qubits of communication offer no computational power beyond BQP. In other words, QRG log = BQP, for QRG log defined appropriately. The details are left to the reader.

Two-message quantum interactive proofs with short answers
In light of the results of Section 3, one may ask if two-message quantum interactive proofs with short answers (as opposed to short questions) have the power of QMA or even BQP. If this is true it is likely to be difficult to show: the graph non-isomorphism problem, which is not known to be in QMA, has a simple and well-known classical protocol [GMW91] requiring polynomial-length questions and constant-length answers. (Indeed, every problem in QSZK has a two-message quantum interactive proof system with a constant-length message from the prover to the verifier, for any choice of constant completeness and soundness errors [Wat02]. ) We can show, however, that public-coin quantum interactive proofs in which the verifier sends polynomially many random bits to the prover, followed by a logarithmic-length quantum message response from the prover, have only the power of BQP.
Following a similar terminology to the classical case, we refer to a quantum interactive proof system in which the verifier's messages to the prover consist of uniformly-generated random bits as quantum Arthur-Merlin games. Let us write QAM([poly, log], a, b) to denote the class of promise problems having two-message quantum Arthur-Merlin games with completeness and soundness probability bounds a and b, in which Merlin's response to Arthur has logarithmic length.
Proof. Assume that B is a promise problem in QAM([poly, log], a, b), and consider a choice of Arthur that witnesses this fact. For r(n) = O(log n), and for any choice of an input string x, Arthur chooses a random string y with length polynomial in | x|, and then measures r = r(| x|) qubits sent by Merlin with respect to some binary-valued measurement {P x,y 0 , P x,y 1 } that depends on x and y. Thus, assuming that the randomly chosen string is y, the maximum acceptance probability of Arthur is equal to the spectral norm of P x,y 1 . So to find P x,y 1 (and its norm) we perform quantum state tomography on the normalized Choi-Jamiołkowski representation of the channel Φ x,y (σ) = P x,y 0 , σ |0 0| + P x,y 1 , σ |1 1| , which describes Arthur's measurement.
The following algorithm shows that B ∈ BQP.
1. Choose y uniformly at random (just as Arthur does).
Prepare N copies of the state ρ, defined to be the normalized Choi-Jamiołkowski representation of Φ x,y , and perform quantum state tomography of ρ. Let H denote the result. Then by Lemma 1 with probability at least 1 − ε, ρ − H 1 ≤ ε.
It can easily be shown that α y is an approximation of P x,y 1 . If α y ≥ 1 then accept. Otherwise, accept with probability α y and reject otherwise.
Since the maximum acceptance probability of Arthur is equal to the expectation value of P x,y 1 over the random choice of y, and with probability 1 − ε, α y is within distance 2 r ε of P x,y 1 , the above procedure has acceptance probability within 1/(4p) of the maximum acceptance probability of Arthur. Therefore B ∈ BQP.

Open problems
Besides the open problems mentioned above, the following two questions have been raised by unknown referees which we leave for future works. The first one is to find the expressive power of the following model: the verifier and prover send messages to each other with the total number of O(log n) qubits and at the end the prover sends a polynomial-size message to the verifier. This model contains QMA and it seems that combining the ideas in Sections 3 and 4 the other direction can also be proved. The other model is an interactive protocol in which the verifier always sends public-coin messages to the prover of the total polynomial-length and the prover replies with qubits of the total logarithmic-length.