[论文解读] Quantum 2-SAT on Low Dimensional Systems Is QMAsubscript{1}-Complete: Direct Embeddings and Black-Box Simulation

Sevag Gharibian, Dorian Rudolph|arXiv (Cornell University)|Jun 10, 2022

Complexity and Algorithms in Graphs被引用 2

一句话总结

该论文通过引入流式量子证明模型（SQCMASPACE）并证明其等于NEXP，确立了低维系统上量子2-SAT问题的QMA₁-完全性。研究表明，当使用指数长度的经典证明时，量子约束满足问题的解空间始终可通过局部酉模拟连通，并展示了如何通过一个QMA(2)-完全的稀疏可分哈密顿量问题，将任意SQCMASPACE计算嵌入无纠缠的证明者中，从而解决了关于基态连通性（Ground State Connectivity）的一个开放问题。

ABSTRACT

A celebrated result in classical complexity theory is Savitch’s theorem, which states that non-deterministic polynomial-space computations (NPSPACE) can be simulated by deterministic poly-space computations (PSPACE). In this work, we initiate the study of a quantum analogue of NPSPACE, denoted Streaming-QCMASPACE (SQCMASPACE), in which an exponentially long classical proof is streamed to a poly-space quantum verifier. We first show that a quantum analogue of Savitch’s theorem is unlikely to hold, in that SQCMASPACE = NEXP. For completeness, we also introduce the companion class Streaming-QMASPACE (SQMASPACE) with an exponentially long streamed quantum proof, and show SQMASPACE = QMAEXP (the quantum analogue of NEXP). Our primary focus, however, is on the study of exponentially long streaming classical proofs, where we next show the following two main results. The first result shows that, in strong contrast to the classical setting, the solution space of a quantum constraint satisfaction problem (i.e. a local Hamiltonian) is always connected when exponentially long proofs are permitted. For this, we show how to simulate any Lipschitz continuous path on the unit hypersphere via a sequence of local unitary gates, at the expense of blowing up the circuit size. This shows that quantum error-correcting codes can be unable to detect one codeword erroneously evolving to another if the evolution happens sufficiently slowly, and answers an open question of [Gharibian, Sikora, ICALP 2015] regarding the Ground State Connectivity problem. Our second main result is that any SQCMASPACE computation can be embedded into "unentanglement", i.e. into a quantum constraint satisfaction problem with unentangled provers. Formally, we show how to embed SQCMASPACE into the Sparse Separable Hamiltonian problem of [Chailloux, Sattath, CCC 2012] (QMA(2)-complete for 1/poly promise gap), at the expense of scaling the promise gap with the streamed proof size. As a corollary, we obtain the first systematic construction for obtaining QMA(2)-type upper bounds on arbitrary multi-prover interactive proof systems, where the QMA(2) promise gap scales exponentially with the number of bits of communication in the interactive proof. Our construction uses a new technique for exploiting unentanglement to simulate quadratic Boolean functions, which in some sense allows history states to encode the future.

研究动机与目标

• 形式化并研究具有指数长度经典证明的NPSPACE的量子类比，引入流式-QCMASPACE（SQCMASPACE）类。
• 探究指数长度证明是否会使原本在多项式长度证明下困难的量子复杂性问题变得平凡化。
• 通过证明在允许指数长度证明时，局部哈密顿量的解空间始终连通，从而解决基态连通性（GSCON）问题。
• 系统性地将多证明者交互式证明嵌入无纠缠量子约束满足问题中，从而获得QMA(2)类型的上界。

提出的方法

• 提出一种流式模型，其中经典证明以逐比特方式输入给一个使用单量子比特门（I或X）作用于指定量子比特的多项式空间量子验证者。
• 开发一种技术，通过一系列局部酉门模拟单位超球面上的任意Lipschitz连续路径，从而实现量子态的连续演化。
• 构建SQCMASPACE到稀疏可分哈密顿量问题的直接嵌入，该问题已被证明是QMA(2)-完全的，且保证间隙与证明长度成反比。
• 利用历史态构造，通过无纠缠实现对未来的演化模拟，从而通过无纠缠证明者模拟二次布尔函数。
• 应用遍历引理（Traversal Lemma）来界定路径上的最小能量，确保在GSCON约化中的可靠性。
• 将PSPACE-完全问题约化为精确的k-LH实例，其保证间隙为反指数级，随后将这些实例嵌入GSCONexp。

实验结果

研究问题

• RQ1 是否存在Savitch定理的量子类比，即SQCMASPACE是否等于PSPACE？
• RQ2 指数长度的经典证明是否总能使量子约束满足问题（如局部哈密顿量）的解空间连通？
• RQ3 任何SQCMASPACE计算是否可嵌入到无纠缠证明者的QMA(2)-完全问题中？
• RQ4 当允许指数长度证明时，基态连通性（GSCON）问题是否变得平凡？
• RQ5 多证明者交互式证明是否可系统性地通过QMA(2)上界表示，且QMA(2)的保证间隙随通信比特数呈指数级变化？

主要发现

• SQCMASPACE等于NEXP，表明Savitch定理的量子类比不成立。
• 当允许指数长度经典证明时，任何局部哈密顿量的解空间始终可通过局部酉门模拟连续路径而连通。
• 任何SQCMASPACE计算可被嵌入到稀疏可分哈密顿量问题中，该问题为QMA(2)-完全问题，且保证间隙按1/poly(2^p(n))缩放，其中p(n)为证明长度。
• 基态连通性问题已解决：在相同证明模型下，其解空间始终连通，回答了Gharibian与Sikora（ICALP 2015）提出的一个开放问题。
• 本文提供了一种系统性构造方法，用于使用QMA(2)对多证明者交互式证明进行上界估计，其中QMA(2)的保证间隙随通信比特数呈指数级缩放。
• 本文确立了SEPARABLE SPARSE GSCONexp为NEXP-完全问题，扩展了基态连通性问题的复杂性分类。