Witness-preserving Amplification of QMA

Transcription

1 Witness-preserving Amplification of QMA Yassine Hamoudi December 30, 2015 Contents 1 Introduction QMA and the amplification problem Prior works Some facts about projectors Jordan s lemma Alternative formulation of QMA The witness-preserving amplification from [MW05] Description of the algorithm Analysis Comments and perspectives 8 5 Conclusion 9 References 9 1 Introduction One of the great questions of quantum computation is to understand how the quantum framework could speed-up classical algorithms. One way to tackle this problem is to define quantum analogues for the classical complexity classes (NP, BPP, IP,... ), and then study how they are related to each other. In this report, we will be interested in the quantum class QMA (Quantum Merlin Arthur), an analogue of NP and MA. This class characterizes the languages that can be recognize in polynomial time by quantum randomized circuits using quantum polynomial size witnesses. Among the famous results associated to MA and BPP is how to decrease the error probability of randomized circuits. Some attempts have been done to transpose this result in the quantum setting. Here we are going to present the amplification algorithm from [MW05] that achieves this task for QMA. In the following of this section, we will describe more formally the class QMA, the related amplification problem and some prior works. Then, some tools about projectors will be presented in section 2. This will enable us to detail in section 3 1

2 a simplified proof of the amplification scheme from [MW05]. Finally, a more recent result, from [NWZ09], will be compared to the last one in section QMA and the amplification problem A family of circuits {V x : x {0, 1} } is uniformly generated if there exists a polynomial time algorithm that given x (in binary) generates V x. This notion is more convenient than Turing machines to define quantum analogues of classical classes. We especially use it for MA and QMA. Definition 1. The (classical) class MA(a, b) consists of all languages L for which there exists a uniformly generated family of classical, randomized, poly-size circuits {V x : x {0, 1} }, and a polynomial m such that: 1. (Completeness) If x L then there exists a witness w of size m( x ) such that P(V x accepts w) a. 2. (Soundness) If x / L then for all witness w of size m( x ), we have P(V x accepts w) b. NP is obtained by removing the randomness in the previous definition, and BPP the witnesses. We also refer to MA(2/3, 1/3) as MA. Finally, QMA is defined using quantum circuits and witnesses: Definition 2. The (quantum) class QMA(a, b) consists of all languages L for which there exists a uniformly generated family of quantum, randomized, poly-size circuits {V x : x {0, 1} }, and two polynomials m, k such that V x has m( x ) input qubits, k( x ) ancillae (auxiliary qubits) and: 1. (Completeness) If x L then there exists a witness φ of size m( x ) such that P(V x accepts φ 0 k( x ) ) a. 2. (Soundness) If x / L then for all witness φ of size m( x ), we have P(V x accepts φ 0 k( x ) ) b. We will also call witnesses the tensor products ψ = φ 0 k( x ), and QMA will refer to QMA(2/3, 1/3). Finally, the quantum analogue BQP of BPP is obtained by removing the witnesses in the previous definition. Figure 1: A QMA verifier V. Note that the output qubit is on the first row. It is interesting to reduce the completeness and soundness errors in the definitions of MA and QMA, both for practical purposes and motivating the conventions MA = MA(2/3, 1/3) and QMA = QMA(2/3, 1/3). The following well-known theorem already provides an exponential decrease for MA: 2

3 Theorem 3. Let a, b and q three polynomials such that a(n) b(n) 1/q(n). Then, for all polynomial r and language L MA(a, b) we have L MA ( 1 2 r( x ), 2 r( x )). The proof is quite obvious. We just need to apply the original verifier polynomially many times, and then take the majority result. Standard Chernoff bound provides the new error term. The quantum version of this problem can be stated as follow: Problem 1. Given a polynomial r and a language L QMA(a, b) recognized by the family of circuits {V x : x {0, 1} }, is it possible to define a new family {V x : x {0, 1} } that recognizes L with completeness probability 1 2 r( x ), and soundness probability 2 r( x ) (i.e. L QMA ( 1 2 r( x ), 2 r( x )) )? Our goal in the following is to present algorithms for solving Problem 1. We will focus on a single quantum randomized poly-size circuit V with completeness probability a, and soundness probability b. The amplification will provide new circuits V that decrease the error of V. These results can then be applied to a whole family {V x : x {0, 1} }, thus solving the previous problem. 1.2 Prior works The complexity class QMA seems to be defined first in [Kit99] and [Wat00]. Various results concerning it have been proved in [KSV02], such as the completeness of the Local Hamiltonian Problem and the first amplification result solving Problem 1: Theorem 4 ([KSV02]). The amplification problem for ( circuit ) V (defined as in Problem 1) can be solved with a circuit V which uses O r copies of the original (a b) ( ) 2 witness, and is made of O r copies of V. (a b) 2 Proof. The idea is to mimic the classical amplification proof, by applying the original verifier several ( times ) and then output the majority result. This needs to run independently O r copies of V, and feed each of them with a new copy of the (a b) 2 original witness. In the case that x L, there exists a witness ψ such that P(V accepts ψ 0 k( x ) ) 2/3. We then give ψ ψ ψ as input to the new verifier V, which will accept with probability greater than 1 2 r( x ) (standard Chernoff bound). If x / L, the same reasoning applies to states of the form ψ ψ ψ, which will be rejected with probability greater than 1 2 r( x ).The only difficulty is to consider witnesses which are highly entangled states (i.e. not factorable into ψ ψ ψ ). This however is proved not to change the soundness probability. The main drawback of previous amplification is the need of multiple identical witnesses, which increases the size of the quantum states that are manipulated. This issue was solved later in two other amplification procedures for QMA. The first one, which will be described in section 3, makes use of only one copy of the original witness. The second one also lowers the number of copies of V needed in V. We will present it briefly in section 4. 3

4 2 Some facts about projectors A projector Π is a Hermitian matrix satisfying Π 2 = Π. In particular, the eigenvalues of Π are 0 or 1, and it projects on the eigenspace corresponding to the eigenvalue 1. Projectors are a convenient way of describing subspaces. Here we study the interaction of two projectors, and how it can be used to reformulate the QMA framework. 2.1 Jordan s lemma Given two projectors Π 1 and Π 2 defined over the Hilbert space H, the Jordan s lemma provides a nice way to decompose H. Lemma 5 ([Jor75]). For any two projectors Π 1 and Π 2, there exists an orthogonal decomposition of the Hilbert space H into: 1. one-dimensional subspaces T j, on which Π 1 and Π 2 are identities or zero-rank projectors. 2. two-dimensional subspaces S i invariant under Π 1 and Π 2, on which Π 1 and Π 2 are rank-one projectors (i.e. inside each S i there are two unit vectors v i and w i such that Π 1 projects on v i and Π 2 projects on w i ). Proof. An easy proof of this lemma can be found in [Reg06]. We are particularly interested in the two-dimensional subspaces S i (we will get rid of the T j in the following). According to Jordan s lemma, Π 1 is the projector v i v i inside S i, and Π 2 is the projector w i w i. If we set carefully the phases of our vectors, we can obtain two unit vectors v i and w i in S i, orthogonal to v i and w i respectively, such that the configuration of Figure 2 happens. Note that v i w i 0, θ i = arccos( v i w i ) [0, π/2] and p i = cos 2 θ i = v i w i 2. Figure 2: Π 1 and Π 2 inside the two-dimensional subspace S i. We especially obtain four useful relations, for each subspace S i : v i = p i w i + 1 p i w i (1) w i = p i v i + 1 p i v i (2) 4

5 v i = 1 p i w i p i w i (3) 2.2 Alternative formulation of QMA w i = 1 p i v i p i v i (4) We now consider a verifier circuit V acting on m qubits and k ancillae. We define the following related projectors: Π 0 = 0 0 I m+k 1 0 = 1 1 I m+k 1 Π 1 = I m 0 k 0 k 1 = I m+k Π 1 Π 2 = V 0 V {Π 0, 0 } describes the measurement of the first qubit, whereas the measurement associated to {Π 1, 1 } returns 1 if and only the k auxiliary qubits are equal to 0 (i.e. they are set to their initial state). We can use these projectors to define the acceptance probability of V. Lemma 6. The maximum acceptance probability of verifier V is the largest eigenvalue of Π 1 Π 2 Π 1. Proof. If state ψ has auxiliary qubits orthogonal to 0 k, then Π 1 ψ = 0, and ψ Π 1 Π 2 Π 1 ψ = 0. Thus, we only have to consider valid input states ψ = φ 0 k, where φ is a witness. For such a state, we have: ψ Π 1 Π 2 Π 1 ψ = ψ V 0 V ψ = 0 V ψ 2 These quantity is both the eigenvalue associated to ψ and the acceptance probability of φ. Thus, the maximum acceptance probability of V is the largest eigenvalue of Π 1 Π 2 Π 1. This result proves that QMA is in fact a largest eigenvalue problem. Note that the Local Hamiltonian Problem, which is complete for the class QMA ([KKR06]), consists in deciding extremal bounds on the smallest eigenvalue of a certain Hamiltonian. Furthermore, the Jordan s lemma allows us to characterize more precisely the maximum acceptance probability. Lemma 7. Using the notations introduced in the Jordan s lemma applied to the previous projectors Π 1 and Π 2, we have: Π 1 Π 2 Π 1 = p i v i v i i=1,2,... In particular, the maximum acceptance probability of V (or similarly its largest eigenvalue) is max i p i. 5

6 Proof. We will assume in the following that the acceptance probability of V for any input is strictly between 0 and 1 (if this is not the case, toss a three-sided dice before running V and either accept, reject, or run V depending on the result). In each one-dimensional subspace T j, if Π 1 is not zero then it is a rank-one projector. So there exists u i in Π 1 (i.e. a legal witness, with auxiliary qubits set to 0) such that Π 1 u i = u i. Then, depending on whether Π 2 is 0 or 1 in T j, we obtain either v i Π 1 Π 2 Π 1 v i = 0 or v i Π 1 Π 2 Π 1 v i = 1. Consequently, v i is accepted with probability 0 or 1, which contradicts the previous assumption. Thus, Π 1 must be zero in each T j, and Π 1 is spanned only by v 1, v 2,.... Moreover, in each S i we have Π 1 Π 2 Π 1 = v i v i w i w i v i v i = v i w i 2 v i v i = p i v i v i. Bringing all the S i together we obtain Π 1 Π 2 Π 1 = p i v i v i. i=1,2,... 3 The witness-preserving amplification from [MW05] C. Marriott and J. Watrous improved the first amplification result from Theorem 4, by removing the need of multiple copies of the original witness. Here is their main result we will detail in the following: Theorem 8 ([MW05]). The amplification problem for circuit V (defined as in Problem( 1) can ) be solved with a circuit V which uses only the original witness and O r copies of V. (a b) Description of the algorithm We consider a verifier V acting on m qubits and k ancillae. We define a new circuit V with same inputs as V, which repeats N times the following two steps: 1. apply V, measure if the output qubit is 1, apply V 2. measure if all auxiliary qubits are 0. We then write down the 2N measurement results and count the number of equal consecutive values. For instance, if the measurements are YYYNYNYYNN then it is 4. Finally, if the count is greater than N a+b 2 then V accepts, otherwise it rejects. Figure 3 illustrates the circuit V (the S box performs the previous counting). Figure 3: The amplified verifier V from [MW05]. 6

7 Thus, verifier V measures alternatively in {Π 2, I Π 2 } and {Π 1, 1 }. Its new completeness and soundness probabilities depend on the number N of steps performed. We set this parameter in the next section. 3.2 Analysis We first look at the computation of V on an input vector v i (using notations of section 2.2). According to Equation 1, we have v i = p i w i + 1 p i wi. However, the first measurement is {Π 2, I Π 2 }, where Π 2 = w i w i. Thus, we obtain w i (i.e. Y) with probability p i, and wi (i.e. N) with probability 1 p i. Then, depending on the state we obtained, measurement {Π 1, 1 } will provide v i or vi with probability p i or 1 p i. In fact, the whole process is a random walk on the Markov chain of Figure 4. Y, Y, Y, N, N, N, Y, Y, Y, N, N, N, Figure 4: Transition probabilities for V on input v i or v i. Starting at any point of this chain, the probability that the next two steps provide equal values (i.e. YY or NN) is p 2 i + p i(1 p i ) = p i. The probability of a given sequence of results is thus p j i (1 p i) N j, where j is the number of equal consecutive values in the sequence. Consequently, the probability of obtaining exactly j equal consecutive values is ( N j ) p j i (1 p i) N j, and V accepts v i with probability: N j=n a+b 2 ( ) N p j i j (1 p i) N j (5) Finally, we obtain the parameter N that makes amplification work, and concludes the proof of Theorem 8: Lemma ( ) 9 ([MW05]). If the two steps of previous algorithm are repeated N = O r times, then V achieves the amplified bounds of Problem 1. (a b) 2 Proof. First, let us assume that x L (i.e. there is a witness ψ such that P(V accepts ψ ) a). According to Lemma 6, this means that the largest eigenvalue of Π 1 Π 2 Π 1 is greater than a. However, the eigenvalues of Π 1 Π 2 Π 1 are exactly the p i s (Lemma 7). Thus, there exists i such that p i a. Using standard Chernoff bound on Equation 5, we conclude that V accepts v i with probability greater than 1 2 r. If x / L, all the p i s are less than b (according to Lemmas 6 and 7). The same reasoning as previously shows the v i s are accepted by V with probability less than 7

8 2 r. Finally, any witness ψ is a linear combination of the v i s, and it is easy to prove that it satisfies the new soundness property 4 Comments and perspectives Theorem 8 removes the multiple witnesses needed in Theorem 4, thus bringing the QMA amplification more practical. For instance, this leads to an analogue of the classical equality MA log = BPP, namely QMA log = BQP (where MA log and QMA log use logarithmic size witnesses only). This result, that easily stems from Theorem 8 (see [Reg06] for instance), could not be obtained with Theorem 4 since it requires to reduce the errors without polynomially increasing the size of the witness. A more recent amplification result, from [NWZ09], is also quadratically faster than Theorem 8: Theorem 10 ([NWZ09]). The amplification problem for circuit V (defined as in Problem ( ) 1) can be solved with a circuit V which uses only the original witness and O r a b copies of V. Proof. The algorithm builds upon [MW05], using the same projectors Π 1 and Π 2. It also considers the reflections about their supports: F 1 = 2Π 1 I F 2 = 2Π 2 I (6) Using the Jordan s lemma, the authors prove that in each two dimensional subspace S i, the product F 1 F 2 is a rotation by an angle ϕ i. Moreover, ϕ i is small if and only if the acceptance probability of original verifier V is high. Thus, the amplification becomes a phase estimation problem. Finally, the procedure is repeated several times to decrease the phase estimation error. There exist even stronger results in the classical case. For instance, it was proved in [ZF87] that the class MA 1, which requires perfect completeness (i.e. if x L then V x accepts a witness with probability 1), was in fact equal to MA. We do not know whether QMA 1 = QMA also. However, the work carried out in [NWZ09] provides QMA 1 = QMA ϕ for a subclass QMA ϕ of QMA. Finally, generalizing the definition of QMA, by allowing Arthur and Merlin to exchanged k messages (instead of only one witness), leads to the definition of QIP(k). In particular, BQP = QIP(0) and QMA = QIP(1). More generally, the quantum analogue QIP (Quantum Interactive Proofs) of IP allows polynomial many rounds between Arthur and Merlin. However, a striking result from [KW00] proves that QIP = QIP(3), which demonstrates that more than three rounds is useless. This is one of the only differences between quantum and classical classes, since IP = IP(3) is highly unlikely (unless AM = PSPACE). Moreover, it was shown recently that QIP = PSPACE ([JJUW11]). Since IP = PSPACE too ([Sha92]), it implies that classical interactive proofs can simulate quantum ones! The most misunderstood class remains QIP(2). 8

9 5 Conclusion We detailed in this report some results concerning the quantum analogue QMA of MA. We especially described three amplification theorems, from [Wat00], [MW05] and [NWZ09], the later two having the advantage of being witness-preserving. Other classical properties, such as complete problems or perfect completeness, have also been mentioned in the quantum setting. All of these results illustrate the growing similarities between classical and quantum computation. Some existing relationships have been summarized Figure 5. BQP QMA QIP(2) QIP = IP = PSPACE BPP MA P NP Figure 5: Some inclusions between classical and quantum classes. However, many problems regarding P and NP have not been solved, or even transposed, in the quantum setting yet (e.g. the PCP theorem). It is also not clear that QMA is the right analogue of NP. Thus, other classes such as QCMA (quantum verifier but classical witness) could carry the analogy further. References [JJUW11] Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay, and John Watrous. QIP = PSPACE. J. ACM, 58(6):30:1 30:27, December [Jor75] [Kit99] [KKR06] [KSV02] [KW00] Camille Jordan. Essai sur la géométrie à n dimensions. Bulletin de la Société Mathématique de France, 3: , Alexei Kitaev. Quantum NP. Talk at AQIP 99: Second Workshop on Algorithms in Quantum Information Processing, DePaul University, January Julia Kempe, Alexei Kitaev, and Oded Regev. The complexity of the local Hamiltonian problem. SIAM J. Comput., 35(5): , May A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi. Classical and Quantum Computation. American Mathematical Society, Boston, MA, USA, Alexei Kitaev and John Watrous. Parallelization, amplification, and exponential time simulation of quantum interactive proof systems. In Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, STOC 00, pages , New York, NY, USA, ACM. 9

10 [MW05] Chris Marriott and John Watrous. Quantum Arthur-Merlin games. Comput. Complex., 14(2): , June [NWZ09] Daniel Nagaj, Pawel Wocjan, and Yong Zhang. Fast amplification of QMA. Quantum Info. Comput., 9(11): , November [Reg06] Oded Regev. Witness-preserving amplification of QMA. University Lecture, fall_2005/ln/qma.pdf. [Sha92] Adi Shamir. IP = PSPACE. J. ACM, 39(4): , October [Wat00] [ZF87] J. Watrous. Succinct quantum proofs for properties of finite groups. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, FOCS 00, pages 537, Washington, DC, USA, IEEE Computer Society. Stathis Zachos and Martin Furer. Probabilistic quantifiers vs. distrustful adversaries. In Proc. Of the Seventh Conference on Foundations of Software Technology and Theoretical Computer Science, pages , London, UK, UK, Springer-Verlag. 10