3-Local Hamiltonian is QMA-complete
Julia Kempe, Oded Regev
Kitaev证明了5-局部哈密顿问题是QMA-完全的。在这里，我们减少了局部性的问题，表明3-本地哈密顿已经是QMA-完全的。
It has been shown by Kitaev that the 5-local Hamiltonian problem is QMA-complete. Here we reduce the locality of the problem by showing that 3-local Hamiltonian is already QMA-complete.
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The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of estimating ground state energies of local Hamiltonians. Perhaps surprisingly, [Ambainis, CCC 2014] showed that the related, but arguably more natural, problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^QMA[log]-complet…
Finding the ground state energy of electrons subject to an external electric field is a fundamental problem in computational chemistry. We prove that this electronic-structure problem, when restricted to a fixed single-particle basis and fixed number of electrons, is QMA-complete. Schuch and Verstraete have shown hardness for the electronic-structure problem with an additional site-specific external magnetic field, but without the restriction to a fixed basis. In their reduction, a local Hamiltonian on qubits is encoded in the site-specific magnetic field. In our reduction, the local Hamiltoni…
The rotation-invariant Hamiltonian problem is QMA$_{\rm EXP}$-complete
本文研究固定任意晶格维数并缩放晶格长度时的旋转不变平移不变局部哈密顿量问题，证明其为 QMA$_{\ m EXP}$ 完全，回答了 Gottesman 与 Irani 的开放问题。
QMA has perfect completeness
本文证明QMA等于QMA_1，即所有量子Merlin-Arthur证明系统可完全完备，仅用Hadamard、Toffoli和X门实现，并推出量子3-SAT的QMA完备性及预言机分离扩展。
We show that finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is QMA-complete. We also show that finding the highest energy of a stoquastic Hamiltonian is QMA-complete and that adiabatic quantum computation using certain excited states of a stoquastic Hamiltonian is universal. We also show that adiabatic evolution in the ground state of a stochastic frustration free Hamiltonian is universal. Our results give a new QMA-complete problem arising in the classical setting of Markov chains, and new adiabatically universal Hamiltonians that arise in many physical systems.
The Local Hamiltonian problem is the problem of estimating the least eigenvalue of a local Hamiltonian, and is complete for the class QMA. The 1D problem on a chain of qubits has heuristics which work well, while the 13-state qudit case has been shown to be QMA-complete. We show that this problem remains QMA-complete when the dimensionality of the qudits is brought down to 8.
In this paper, we study variants of the canonical Local-Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local-Hamiltonian problem. The input for the Separable-Local-Hamiltonian problem is the same as the Local-Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable-Sparse-Hamiltonian problem is defined similar…
We show that the two-dimensional (2D) local Hamiltonian problem with the constraint that the ground state obeys area laws is QMA-complete. We also prove similar results in 2D translation-invariant systems and for the 3D Heisenberg and Hubbard models with local magnetic fields. Consequently, unless MA = QMA, not all ground states of 2D local Hamiltonians with area laws have efficient classical representations that support efficient computation of local expectation values. In the future, even if area laws are proved for ground states of 2D gapped systems, the computational complexity of these sy…
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.
Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. We study the natural special case of the Local Hamiltonian problem where the same 2-local interaction, with differing weights, is applied across each pair of qubits. First we consider antiferromagnetic/ferromagnetic interactions, where the weights of the terms in the Hamiltonian are restricted to all be of the same sign. We show that for symmetric 2-local interactions with no 1-local part, the problem is either QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenbe…
The Commuting Local Hamiltonian Problem: Relativized Evidence Against BQP-Hardness
本文通过构造经典预言机，证明BQP类不包含于QIMA类，为一般可交换局域哈密顿问题非BQP困难及非QMA完备提供了相对化证据。
A new construction for a QMA complete 3-local Hamiltonian
提出了一种将量子计算编码为3-定域哈密顿量的新方法。我们的构造是新颖的，因为它不包括任何引起合法-非法时钟转换的术语。因此，哈密尔顿算子中的项的权重不像以前的构造那样与问题的大小成比例。这改进了Kempe和Regev的构造，他们是第一个证明3-局部哈密顿量对于复杂性类QMA（NP的量子类似物）是完备的。量子k-SAT是局部哈密顿问题的一个限制性版本，仅使用投影项，由Bravyi引入作为经典k-SAT问题的模拟。Bravyi证明了量子4-SAT对于具有单侧误差的QMA类（QMA_1）是完备的，量子2-SAT在P中.我们给出了一个量子线路到量子4-SAT哈密顿量的编码.作为这个3-局域构造的中间步骤，我们证明了对于尺寸为3x 2x2（一个qutrit和两个qubit）的粒子的量子3-SAT是QMA_1完备的。量子3-SAT与量子比特的复杂性仍然是一个悬而未决的问题。
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).
The Complexity of the Local Hamiltonian Problem
k-局部哈密顿问题是复杂性类QMA（NP的量子模拟）的自然完备问题。它在精神上类似于MAX-k-SAT，对于k<=2，它是NP完全的。已知对于任何k <= 3，该问题是QMA完全的。另一方面，1-局部哈密顿量在P中，因此不被认为是QMA完备的。2-局部哈密顿问题的复杂性一直是一个突出的问题。在这里，我们解决了这个问题，并表明它是QMA完全的。我们提供了两个独立的证明;我们的第一个证明只使用初等线性代数。我们的第二个证明使用了一个强大的技术来分析两个哈密顿量的总和;这种技术是基于微扰理论，我们相信它可能会证明在其他地方有用。使用我们的技术，我们还表明，绝热计算与两个本地相互作用的量子比特是等效的标准量子计算。
Towards a universal gateset for $\mathsf{QMA}_1$
本文研究 $\mathsf{QMA}_1$ 的通用门集问题，证明分圆域中的门集 $G_{2^k}$ 具有通用性，并给出量子 $l$-SAT、$2$-局部哈密顿量及 Clique Homology 等问题的 $\mathsf{QMA}_1$ 完全性结果。
Zero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are $\mathrm{QMA}_1$-Complete
该研究证明了链上超对称哈密顿量的零能问题属于QMA₁完全，通过一维无挫折哈密顿量构造得到基态能量的逆多项式下界，明确了问题的计算复杂性。
On the complexity of unique quantum witnesses and quantum approximate counting
本文通过量子预言机分离排除了从 QMA 到 UniqueQMA 的黑盒归约，并证明满足本征态热化假说的局部哈密顿量基态能量可通过 UniqueQMA 协议估计，揭示了混沌哈密顿量可能比一般哈密顿量计算更简单。
On the complexity of estimating ground state entanglement and free energy
研究局部哈密顿量基态纠缠估计的复杂性，发现qq-QAM和QMA(2)类在其中起关键作用，并首次给出基于局部哈密顿量的QMA(2)-完全问题。
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…
Two bases suffice for QMA1-completeness
本文提出仅允许标准基与 Hadamard 基对角项的 Quantum-k-SAT 变体，证明受基限制的 Quantum-6-SAT 关于自然门集已是 QMA1-complete，并借助交错时钟构造为量子 PCP 研究提供类 CSS Hamiltonian。
Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem and initiate its study. We present two main results. The first shows that a non-trivial approximation ratio can be obtained in the class NP using product states. The second result (which builds on the first one), gives a polynomial time (classical) algorithm providing a similar approximation ratio for dense instances of the problem. The latter result is based on an…
The Bose-Hubbard model is a system of interacting bosons that live on the vertices of a graph. The particles can move between adjacent vertices and experience a repulsive on-site interaction. The Hamiltonian is determined by a choice of graph that specifies the geometry in which the particles move and interact. We prove that approximating the ground energy of the Bose-Hubbard model on a graph at fixed particle number is QMA-complete. In our QMA-hardness proof, we encode the history of an n-qubit computation in the subspace with at most one particle per site (i.e., hard-core bosons). This featu…
Identity check is QMA-complete
我们定义问题的身份检查：给定一个量子线路的经典描述，确定它是否几乎等价于身份。明确地说，任务是决定相应的酉是否接近于关于算子范数的单位矩阵的复数倍。我们证明了这个问题是QMA完全的。这个问题的一个推广是等价性检验：给定量子线路的两种描述和一个公共不变子空间的描述，判断电路对这个子空间的约束是否几乎一致。我们表明，等价性检查也在QMA，因此QMA完全。
On Perfect Completeness for QMA
QMA类是否等于QMA 1，或具有单侧误差的QMA，多年来一直是一个悬而未决的问题。这篇笔记有助于解释为什么这个问题很难，通过使用来自实分析的想法来给出一个相对于它们不同的“量子预言”。作为一个副产品，我们发现，有一些关于量子复杂性类的事实是经典相对化的，但不是量子相对化的，其中包括ZQEXP中的BQP这样的“平凡”包含。
N-representability is QMA-complete
研究了量子化学中N-可表示性问题的计算复杂性。我们证明了这个问题是QMA-完全的，这是NP-完全的量子推广。我们的证明使用一个简单的映射从自旋系统的费米子系统，以及凸优化技术，减少了寻找基态的问题，N-表示。
The Non-Identity Check problem asks whether a given a quantum circuit is far away from the identity or not. It is well known that this problem is QMA-Complete \cite{JWB05}. In this note, it is shown that the Non-Identity Check problem remains QMA-Complete for circuits of short depth. Specifically, we prove that for constant depth quantum circuit in which each gate is given to at least $Ω(\log n)$ bits of precision, the Non-Identity Check problem is QMA-Complete. It also follows that the hardness of the problem remains for polylogarithmic depth circuit consisting of only gates from any universa…
We prove that 2-Local Hamiltonian (2-LH) with Low Complexity problem is QCMA-complete by combining the results from the QMA-completeness[4] of 2-LH and QCMA-completeness of 3-LH with Low Complexity[6]. The idea is straightforward. It has been known that 2-LH is QMA-complete. By putting a low complexity constraint on the input state, we make the problem QCMA. Finally, we use similar arguments as in [4] to show that all QCMA problems can be reduced to our proposed problem.
On the Pure Quantum Polynomial Hierarchy and Quantified Hamiltonian Complexity
本文证明了纯量子多项式层级的新包含关系、误差约简及与QPH的等价性，并首次研究量化哈密顿量复杂性，证明量化纯稀疏哈密顿量问题是pureQSigma完全的。
We present a classical interactive protocol that verifies the validity of a quantum witness state for the local Hamiltonian problem. It follows from this protocol that approximating the non-local value of a multi-player one-round game to inverse polynomial precision is QMA-hard. Our work makes an interesting connection between the theory of QMA-completeness and Hamiltonian complexity on one hand and the study of non-local games and Bell inequalities on the other.
Cooling and Low Energy State Preparation for 3-local Hamiltonians are FQMA-complete
我们引入了量子复杂性类ARMA。这门课描述了生成一个量子态的复杂性，这个量子态可以作为给定QMA问题的证明。从某种意义上说，FNP（与NP相关的函数问题）的量子类比。后者描述了为NP决策问题找到简洁证明的复杂性。而所有的FNP问题可以减少到NP，有没有明显的减少QMA，因为QMA的解决方案是一个量子态和QMA的解决方案的答案是或否。我们认为量子态生成器，得到经典的描述3-本地哈密顿量的n量子位作为输入，并准备这些系统的低能量状态作为输出。我们表明，这种状态发生器可以用来准备证人QMA问题。因此，低能态制备是完全的。我们的证明是Kitaev等人和Kempe与Regev关于k-局部Hamilton问题的QMA-完备性证明的推广。我们表明，可以解决的准备热平衡态的适当的温度下降作为一个多项式的倒数n。
A QMA-Complete Translationally Invariant Hamiltonian Problem and the Complexity of Finding Ground State Energies in Physical Systems
在这里，我们提出了一个问题有关的局部哈密顿问题（确定是否基态能量属于两个范围之一），这是限制为不变性。我们证明了对于具有固定的局部维数和O（\log（N））-体局部项的问题，或具有局部维数N和2-体项的问题，存在问题是QMA-完全的例子.我们讨论的计算复杂性，找到这些系统的基态的影响，因此，任何经典的近似技术，可以适用于包括DMRG，矩阵乘积状态和MERA。一个重要的例子是在恒定填充分数下最近邻跳跃的玻色子的一维晶格，即玻色-哈伯德模型的推广。
The quantum PCP (QPCP) conjecture states that all problems in QMA, the quantum analogue of NP, admit quantum verifiers that only act on a constant number of qubits of a polynomial size quantum proof and have a constant gap between completeness and soundness. Despite an impressive body of work trying to prove or disprove the quantum PCP conjecture, it still remains widely open. The above-mentioned proof verification statement has also been shown equivalent to the QMA-completeness of the Local Hamiltonian problem with constant relative gap. Nevertheless, unlike in the classical case, no equivale…
What could happen if we pinned a single qubit of a system and fixed it in a particular state? First, we show that this can greatly increase the complexity of static questions -- ground state properties of local Hamiltonian problems with restricted types of terms. In particular, we show that the Pinned commuting and Pinned Stoquastic Local Hamiltonian problems are QMA complete. Second, we show that pinning a single qubit via often repeated measurements also results in universal quantum computation already with commuting and stoquastic Hamiltonians. Finally, we discuss variants of the Ground Sta…
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.
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 prove a deterministic exponential time upper bound for Quantum Merlin-Arthur games with k unentangled provers. This is the first non-trivial upper bound of QMA(k) better than NEXP and can be considered an exponential improvement, unless EXP=NEXP. The key ideas of our proof are to use perturbation theory to reduce the QMA(2)-complete Separable Sparse Hamiltonian problem to a variant of the Separable Local Hamiltonian problem with an exponentially small promise gap, and then to decide this instance using epsilon-net methods. Our results imply an exponential time algorithm for the Pure State N…
Two QCMA-complete problems
QMA和QCMA是复杂性类NP的可能量子类似物。在QCMA中，验证器是一个量子程序，证明是经典的。相反，在QMA中，证明也是一个量子态。我们证明了两个已知的QMA-完全问题可以自然地转化为QCMA-完全问题：（1）判定一个3-局部哈密顿量是否具有低能态（其能量小于给定值），并且该低能态最多可以用k个基本门来制备是QCMA-完全的，而当对制备复杂性的限制被取消时，它是QMA-完全的. (2)决定一个（经典描述的）量子线路是否在所有基态上几乎都是同一的，是QCMA完备的。它是QMA-完全的，决定它是否几乎作为单位元作用于所有状态。
The Fermionic Cohomology Problem on the Full Fock Space Is $\mathrm{QMA}_1$-Complete
该研究证明全福克空间上的费米子上同调问题为QMA₁完全，配套证明指定次数的对应问题也为QMA₁完全，为相关量子复杂度问题提供了关键结论。
Consistency of Local Density Matrices is QMA-complete
假设我们有一个n量子比特系统，我们被给定一个局部密度矩阵rho_1，.，rho_m，其中每个rho_i描述量子位的子集C_i。我们说rho_i是"一致的“，如果存在一些全局状态sigma（在所有n个量子比特上）与子集C_i上的每个rho_i匹配。这推广了边际概率分布一致性的经典概念。证明了局部密度矩阵的一致性判定是QMA完全的（其中QMA是NP的量子模拟）。这给出了QMA中一个很难的问题的有趣例子。我们的证明有点不寻常：本文利用Bertsimas和Vempala提出的基于随机抽样的凸优化算法，从局部哈密顿量出发，给出了一个图灵约化。与经典情况不同，简单的映射约简在这里似乎不起作用。
A Note on the Complexity of the Spectral Gap Problem
本文证明了谱隙问题在多一归约下是 QMA 困难的，从而得出其在真值表归约下是 $P^{QMA[log]}$ 完全的，并推测其在多一归约下属于更严格的子类。
New Approaches to Complexity via Quantum Graphs
本文引入量子图团问题，利用量子信道联系，证明其在不同信道族上分别对QMA(2)、QMA、MA和NP完备，首次实现四类直接比较，并给出自测试启发的直接证明及QMA(k)到QMA(2)的新归约。
The Local Consistency Problem for Stoquastic and 1-D Quantum Systems
局域哈密顿问题（求量子系统的基态能量）是量子力学中的一个完备问题。局部一致性问题（决定量子系统的小块描述是否一致）也被称为QMA完全问题。在这里，我们考虑局部哈密顿量的特殊情况，对于“随机”和1维系统，这似乎比QMA更容易。我们表明，存在类似的特殊情况下，本地一致性，具有同等的复杂性（多时间的预言减少）。我们的主要技术工具是一个新的减少，从本地一致性到本地哈密尔顿，使用SDP对偶。
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…
We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the dist…
Realizable Hamiltonians for Universal Adiabatic Quantum Computers
已经建立了局域晶格自旋哈密顿量可以用于通用绝热量子计算。然而，在这些证明中使用的2-局部模型哈密顿量是通用的，因此不限制自旋之间所需的相互作用的类型。为了解决这个问题，本文提供了两个简单的模型哈密顿量，是对实现通用绝热量子计算机的实验工作者的实际利益。所提出的模型哈密顿量是已知的最简单的QMA-完全2-局部哈密顿量。具有1-局域横场的2-局域伊辛模型可能是最简单的量子自旋模型，但不太可能是绝热量子计算的通用模型。我们证明，这个模型可以呈现出普遍性和QMA完成通过添加一个可调的2本地横向XX耦合。我们还证明了只有1-局域Z和X场以及2-局域ZX相互作用的自旋模型的普适性和QMA完备性。
Quantum 2-SAT on low dimensional systems is $\mathsf{QMA}_1$-complete: Direct embeddings and black-box simulation
研究量子可满足性复杂性转变，证明低维 $(2,5)$-QSAT 与 1D $(3,d)$-QSAT 是 $\mathsf{QMA}_1$-完全或难的，并引入直接嵌入与黑盒模拟方法。
This paper presents stronger methods of achieving perfect completeness in quantum interactive proofs. First, it is proved that any problem in QMA has a two-message quantum interactive proof system of perfect completeness with constant soundness error, where the verifier has only to send a constant number of halves of EPR pairs. This in particular implies that the class QMA is necessarily included by the class QIP_1(2) of problems having two-message quantum interactive proofs of perfect completeness, which gives the first nontrivial upper bound for QMA in terms of quantum interactive proofs. It…
Public-Key Quantum Authentication and Digital Signature Schemes Based on the QMA-Complete Problem
提出基于QMA-完全问题的公钥量子认证与数字签名协议，无需可信第三方，安全性经严格证明，并分析验证效率与实现路径。
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 $\…
QMA (Quantum Merlin Arthur) is the class of problems which, though potentially hard to solve, have a quantum solution which can be verified efficiently using a quantum computer. It thus forms a natural quantum version of the classical complexity class NP (and its probabilistic variant MA, Merlin-Arthur games), where the verifier has only classical computational resources. In this paper, we study what happens when we restrict the quantum resources of the verifier to the bare minimum: individual measurements on single qubits received as they come, one-by-one. We find that despite this grave rest…
Unweighted Gapped Clique Homology is $\mathsf{QMA}_1$-complete
该研究证明无权间隙团同调是$\textsf{QMA}_1^{g_2}$完全问题，表明间隙承诺而非顶点加权是间隙团同调复杂性的来源。
Quantum PCPs: on Adaptivity, Multiple Provers and Reductions to Local Hamiltonians
定义了涵盖自适应性与多证明者的量子PCP一般形式，通过到局部哈密顿量的归约证明非自适应可模拟自适应PCP，并揭示了量子PCP猜想需要非相对化技术。
Two remarks on the local Hamiltonian problem
本文给出了局部Hamilton问题在Karp约化下的两个自然限制，它们是BQP-完全的。先前已证明QCMA、QMA_1和MA的完整限制。
On the Complexity of Pure-State Consistency of Local Density Matrices
本文研究局部密度矩阵纯态一致性问题（PureCLDM）的复杂度，定义新复杂度类PureSuperQMA并证明PureCLDM和纯N-可表示性对其完全，同时利用代数几何方法将PureCLDM的上界改进为PSPACE，为PureCLDM可能不是QMA(2)完全提供了证据。
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…
A central result in the study of Quantum Hamiltonian Complexity is that the k-Local hamiltonian problem is QMA-complete. In that problem, we must decide if the lowest eigenvalue of a Hamiltonian is bounded below some value, or above another, promised one of these is true. Given the ground state of the Hamiltonian, a quantum computer can determine this question, even if the ground state itself may not be efficiently quantum preparable. Kitaev's proof of QMA-completeness encodes a unitary quantum circuit in QMA into the ground space of a Hamiltonian. However, we now have quantum computing models…
Finding quantum partial assignments by search-to-decision reductions
针对量子搜索到判定归约问题，提出一种经典多项式时间算法，通过 $\mathsf{QMA}$ 预言机输出近最优量子证据的部分赋值（约化密度矩阵）近似，基于电路到哈密顿量映射和低能密度矩阵验证问题实现。
The Complexity of the Consistency and N-representability Problems for Quantum States
QMA（Quantum Merlin-Arthur）是NP类的量子类似物。有几个QMA完全问题，最著名的是Kitaev引入的“局部哈密顿”问题。在这篇论文中，我们展示了一些新的QMA-完全问题。第一个是“局部密度矩阵的一致性”：给定几个密度矩阵，描述一个n量子比特系统的不同（恒定大小）子集，决定这些是否与一个全局状态一致。这个问题最早是由Aharonov提出的。我们表明，它是QMA-完全的，通过一个甲骨文减少从本地哈密顿。这使用的算法凸优化的成员预言，由于Yudin和Nemirovskii。接下来，我们证明量子化学中的两个问题，“费米局域哈密顿”和“N-可表示性”是QMA完全的。这些问题在计算分子体系的基态能量时出现。N-可表示性是最近开发的使用收缩薛定谔方程的数值方法的关键组成部分。虽然这些问题从20世纪60年代就开始研究了，但直到最近量子计算理论才使我们能够正确地描述它们的复杂性。最后，我们研究了一些特殊情况下的一致性问题，有关1维和“随机”系统。我们还给出了另一种证明的结果，由于Jaynes：只要本地密度矩阵是一致的，他们是一致的吉布斯状态。
All Hamiltonian complexity results to date have been proven by constructing a local Hamiltonian whose ground state -- or at least some low-energy state -- is a "computational history state", encoding a quantum computation as a superposition over the history of the computation. We prove that all history-state Hamiltonians must be critical. More precisely, for any circuit-to-Hamiltonian mapping that maps quantum circuits to local Hamiltonians with low-energy history states, there is an increasing sequence of circuits that maps to a growing sequence of Hamiltonians with spectral gap closing at le…
The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the foll…
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