We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3-SAT, namely local Hamiltonians, is QMA complete. The result builds upon the classical Cook-Levin proof of the NP completeness of SAT, but differs from it in several fundamental ways, which we highlight. This result raises a rich array of open problems related to quantum complexity, algorithms and entanglement, which we state at the end of this survey. This survey is the extension of lecture notes taken by Naveh for Aharonov's quantum computation course, held in Tel Aviv University, 2001.
A quantum oracle separation between QMA(2) and QMA
Simons Institute & UC Berkeley（西蒙斯研究所与加州大学伯克利分校）；IBM Research, Columbia University（IBM 研究部，哥伦比亚大学）；Ruhr-University Bochum（波鸿鲁尔大学）；University of California, Berkeley（加州大学伯克利分校）；MIT（麻省理工学院）；University of Washington（华盛顿大学）
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a "quantum oracle" relative to which they are different. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such "trivial" containments as BQP in ZQEXP.
A quantum expander is a unital quantum channel that is rapidly mixing, has only a few Kraus operators, and can be implemented efficiently on a quantum computer. We consider the problem of estimating the mixing time (i.e., the spectral gap) of a quantum expander. We show that this problem is co-QMA-complete. This has applications to testing randomized constructions of quantum expanders, and studying thermalization of open quantum systems.
Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve $\NP$-complete problems given a "short" quantum proof; more precisely, $\NP\subseteq \QMA_{\log}(2)$ where $\QMA_{\log}(2)$ denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion $\NP\subseteq \QMA_{\log}(2)$ has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap $\…
We present a new way of encoding a quantum computation into a 3-local Hamiltonian. Our construction is novel in that it does not include any terms that induce legal-illegal clock transitions. Therefore, the weights of the terms in the Hamiltonian do not scale with the size of the problem as in previous constructions. This improves the construction by Kempe and Regev, who were the first to prove that 3-local Hamiltonian is complete for the complexity class QMA, the quantum analogue of NP. Quantum k-SAT, a restricted version of the local Hamiltonian problem using only projector terms, was introd…
In this paper we give an overview of the quantum computational complexity class QMA and a description of known QMA-complete problems to date. Such problems are believed to be difficult to solve, even with a quantum computer, but have the property that if a purported solution to the problem is given, a quantum computer would easily be able to verify whether it is correct. An attempt has been made to make this paper as self-contained as possible so that it can be accessible to computer scientists, physicists, mathematicians, and quantum chemists. Problems of interest to all of these professions …
We study the computational complexity of the N-representability problem in quantum chemistry. We show that this problem is QMA-complete, which is the quantum generalization of NP-complete. Our proof uses a simple mapping from spin systems to fermionic systems, as well as a convex optimization technique that reduces the problem of finding ground states to N-representability.
Given a verifier circuit for a problem in QMA, we show how to exponentially amplify the gap between its acceptance probabilities in the `yes' and `no' cases, with a method that is quadratically faster than the procedure given by Marriott and Watrous. Our construction is natively quantum, based on the analogy of a product of two reflections and a quantum walk. Second, in some special cases we show how to amplify the acceptance probability for good witnesses to 1, making a step towards the proof that QMA with one-sided error is equal to QMA. Finally, we simplify the filter-state method to search…
The complexity class QMA is the quantum analog of the classical complexity class NP. The functional analogs of NP and QMA, called functional NP (FNP) and functional QMA (FQMA), consist in either outputting a (classical or quantum) witness, or outputting NO if there does not exist a witness.The classical complexity class Total Functional NP (TFNP) is the subset of FNP for which it can be shown that the NO outcome never occurs. TFNP includes many natural and important problems. In the present work we introduce the complexity class of Total Functional QMA (TFQMA), the quantum analog of TFNP. We s…
We show that computational problem of testing the behaviour of quantum circuits is hard for the class of problems known as QMA that can be verified efficiently with a quantum computer. This result is a generalization of the techniques previously used to prove the hardness of other problem on quantum circuits. We use this result to show the QMA-hardness of a weak version of the problem of detecting the insecurity of a symmetric-key quantum encryption system, or alternately the problem of determining when a quantum channel is not private. We also give a QMA protocol for the problem of detecting …
The class MA consists of languages that can be efficiently verified by classical probabilistic verifiers using a single classical certificate, and the class QMA consists of languages that can be efficiently verified by quantum verifiers using a single quantum certificate. Suppose that a verifier receives not only one but multiple certificates. In the classical setting, it is obvious that a classical verifier with multiple classical certificates is essentially the same with the one with a single classical certificate. However, in the quantum setting where a quantum verifier is given a set of qu…
Quantum satisfiability is a constraint satisfaction problem that generalizes classical boolean satisfiability. In the quantum k-SAT problem, each constraint is specified by a k-local projector and is satisfied by any state in its nullspace. Bravyi showed that quantum 2-SAT can be solved efficiently on a classical computer and that quantum k-SAT with k greater than or equal to 4 is QMA1-complete. Quantum 3-SAT was known to be contained in QMA1, but its computational hardness was unknown until now. We prove that quantum 3-SAT is QMA1-hard, and therefore complete for this complexity class.
QMA and QCMA are possible quantum analogues of the complexity class NP. In QCMA the verifier is a quantum program and the proof is classical. In contrast, in QMA the proof is also a quantum state. We show that two known QMA-complete problems can be modified to QCMA-complete problems in a natural way: (1) Deciding whether a 3-local Hamiltonian has low energy states (with energy smaller than a given value) that can be prepared with at most k elementary gates is QCMA-complete, whereas it is QMA-complete when the restriction on the complexity of preparation is dropped. (2) Deciding whether a (clas…
We show that every language in QMA admits a classical-verifier, quantum-prover zero-knowledge argument system which is sound against quantum polynomial-time provers and zero-knowledge for classical (and quantum) polynomial-time verifiers. The protocol builds upon two recent results: a computational zero-knowledge proof system for languages in QMA, with a quantum verifier, introduced by Broadbent et al. (FOCS 2016), and an argument system for languages in QMA, with a classical verifier, introduced by Mahadev (FOCS 2018).
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given …
The class QMA plays a fundamental role in quantum complexity theory and it has found surprising connections to condensed matter physics and in particular in the study of the minimum energy of quantum systems. In this paper, we further investigate the class QMA and its related class QCMA by asking what makes quantum witnesses potentially more powerful than classical ones. We provide a definition of a new class, SQMA, where we restrict the possible quantum witnesses to the "simpler" subset states, i.e. a uniform superposition over the elements of a subset of n-bit strings. Surprisingly, we prove…
We consider coGapSVP_\sqrt{n}, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM\cap coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first non-trivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+. Working with the QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the s…
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a qu…
This paper gives a QMA (Quantum Merlin-Arthur) protocol for 3-SAT with two logarithmic-size quantum proofs (that are not entangled with each other) such that the gap between the completeness and the soundness is Omega(1/n polylog(n)). This improves the best completeness/soundness gaps known for NP-complete problems in this setting.
Determining the worst-case uncertainty added by a quantum circuit is shown to be computationally intractable. This is the problem of detecting when a quantum channel implemented as a circuit is close to a linear isometry, and it is shown to be complete for the complexity class QMA of verifiable quantum computation. This is done by relating the problem of detecting when a channel is close to an isometry to the problem of determining how mixed the output of the channel can be when the input is a pure state. How mixed the output of the channel is can be detected by a protocol making use of the sw…
This paper introduces a new technique for removing existential quantifiers over quantum states. Using this technique, we show that there is no way to pack an exponential number of bits into a polynomial-size quantum state, in such a way that the value of any one of those bits can later be proven with the help of a polynomial-size quantum witness. We also show that any problem in QMA with polynomial-size quantum advice, is also in PSPACE with polynomial-size classical advice. This builds on our earlier result that BQP/qpoly is contained in PP/poly, and offers an intriguing counterpoint to the r…
We study the complexity of computational problems from quantum physics. Typically, they are studied using the complexity class QMA (quantum counterpart of NP) but some natural computational problems appear to be slightly harder than QMA. We introduce new complexity classes consisting of problems that are solvable with a small number of queries to a QMA oracle and use these complexity classes to quantify the complexity of several natural computational problems (for example, the complexity of estimating the spectral gap of a Hamiltonian).
Computing the ground-state energy of interacting electron (fermion) problems has recently been shown to be hard for QMA, a quantum analogue of the complexity class NP. Fermionic problems are usually hard, a phenomenon widely attributed to the so-called sign problem occurring in Quantum Monte Carlo simulations. The corresponding bosonic problems are, according to conventional wisdom, tractable. Here, we discuss the complexity of interacting boson problems and show that they are also QMA-hard. In addition, we show that the bosonic version of the so-called N-representability problem is QMA-comple…
We introduce the quantum complexity class FQMA. This class describes the complexity of generating a quantum state that serves as a witness for a given QMA problem. In a certain sense, FQMA is the quantum analogue of FNP (function problems associated with NP). The latter describes the complexity of finding a succinct proof for a NP decision problem. Whereas all FNP problems can be reduced to NP, there is no obvious reduction of FQMA to QMA since the solution of FQMA is a quantum state and the solution of QMA the answer yes or no. We consider quantum state generators that get classical descripti…
The Local Hamiltonian problem (finding the ground state energy of a quantum system) is known to be QMA-complete. The Local Consistency problem (deciding whether descriptions of small pieces of a quantum system are consistent) is also known to be QMA-complete. Here we consider special cases of Local Hamiltonian, for ``stoquastic'' and 1-dimensional systems, that seem to be strictly easier than QMA. We show that there exist analogous special cases of Local Consistency, that have equivalent complexity (up to poly-time oracle reductions). Our main technical tool is a new reduction from Local Consi…
We provide several advances to the understanding of the class of Quantum Merlin-Arthur proof systems (QMA), the quantum analogue of NP. Our central contribution is proving a longstanding conjecture that the Consistency of Local Density Matrices (CLDM) problem is QMA-hard under Karp reductions. The input of CLDM consists of local reduced density matrices on sets of at most k qubits, and the problem asks if there is an n-qubit global quantum state that is consistent with all of the k-qubit local density matrices. The containment of this problem in QMA and the QMA-hardness under Turing reductions…
We initiate the study of non-interactive zero-knowledge (NIZK) arguments for languages in QMA. Our first main result is the following: if Learning With Errors (LWE) is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely cla…
QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most notably the ``Local Hamiltonian'' problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems. The first one is ``Consistency of Local Density Matrices'': given several density matrices describing different (constant-size) subsets of an n-qubit system, decide whether these are consistent with a single global state. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. This uses alg…
We give a simpler proof of one of the results of Kobayashi, Le Gall, and Nishimura [arXiv:1210.1290v2], which shows that any QMA protocol can be converted to a one-sided error protocol, in which Arthur and Merlin initially share a constant number of EPR pairs and then Merlin sends his proof to Arthur. Our protocol is similar but somewhat simpler than the original. Our main contribution is a simpler and more direct analysis of the soundness property that uses well-known results in quantum information such as properties of the trace distance and the fidelity, and the quantum de Finetti theorem.
We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property $Π$ and reject inputs that are $\varepsilon$-far from $Π$, while only probing a minuscule portion of their input. We investigate the complexity landscape of this model, showing that QMAPs can be exponentially stronger than both classical proofs of proximity and quantum testers. To this end, we extend the m…
We propose three constructions of classically verifiable non-interactive zero-knowledge proofs and arguments (CV-NIZK) for QMA in various preprocessing models. - We construct a CV-NIZK for QMA in the quantum secret parameter model where a trusted setup sends a quantum proving key to the prover and a classical verification key to the verifier. It is information theoretically sound and zero-knowledge. - Assuming the quantum hardness of the learning with errors problem, we construct a CV-NIZK for QMA in a model where a trusted party generates a CRS and the verifier sends an instance-independent q…
We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly MQA (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class BellQMA(poly), characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resul…
An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity class P^QMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is P^QMA[log]-complete. In this paper, we continue the study of P^QMA[log], obtaining the following lower and upper bounds. Lower bounds (hardness results): (1) The P^QMA[log]-completeness result of [Ambainis, CCC 2014] requires O(log n)-local obse…
In known constructions of classical zero-knowledge protocols for NP, either of zero-knowledge or soundness holds only against computationally bounded adversaries. Indeed, achieving both statistical zero-knowledge and statistical soundness at the same time with classical verifier is impossible for NP unless the polynomial-time hierarchy collapses, and it is also believed to be impossible even with a quantum verifier. In this work, we introduce a novel compromise, which we call the certified everlasting zero-knowledge proof for QMA. It is a computational zero-knowledge proof for QMA, but the ver…
We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is QMA-hard, under randomized reductions, to distinguish between the cases when the players have a strategy using entanglement that succeeds with probability 1 in the game, or when no such strategy succeeds with probability larger than 1/2. This proves the "games quantum PCP conjecture" of Fitzsimons and the second author (ITCS'15), albeit under randomized reductions. The core component in our reduction is a construction of a family of two-player games for testing $n…
It has been established that local lattice spin Hamiltonians can be used for universal adiabatic quantum computation. However, the 2-local model Hamiltonians used in these proofs are general and hence do not limit the types of interactions required between spins. To address this concern, the present paper provides two simple model Hamiltonians that are of practical interest to experimentalists working towards the realization of a universal adiabatic quantum computer. The model Hamiltonians presented are the simplest known QMA-complete 2-local Hamiltonians. The 2-local Ising model with 1-local …
The standard inputs given to a quantum machine are classical binary strings. In this view, any quantum complexity class is a collection of subsets of $\{0,1\}^{*}$. However, a quantum machine can also accept quantum states as its input. T. Yamakami has introduced a general framework for quantum operators and inputs \cite{Yam02}. In this paper we present several quantum languages within this model and by generalizing the complexity classes QMA and QCMA we analyze the complexity of the introduced languages. We also discuss how to derive a classical language from a given quantum language and as a…
