Savitch's theorem states that NPSPACE computations can be simulated in PSPACE. We initiate the study of a quantum analogue of NPSPACE, denoted Streaming-QCMASPACE (SQCMASPACE), where an exponentially long classical proof is streamed to a poly-space quantum verifier. Besides two main results, we also show that a quantum analogue of Savitch's theorem is unlikely to hold, as SQCMASPACE=NEXP. For completeness, we introduce Streaming-QMASPACE (SQMASPACE) with an exponentially long streamed quantum proof, and show SQMASPACE=QMA_EXP (quantum analogue of NEXP). Our first main result shows, in 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 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.
翻译:鼠标SAPACE(SQCASPAACE) 的理论显示, SURPACE 的量类比(SQCASPACE) 无法在 PSPACE (SQCASPACE) 中模拟 。 我们开始研究一个量子类比( PURPACE ), 以极长的量类比( SURPACE ) 来模拟 PSPACE (SQCASPACE ) 。 我们的首个主要结果显示, 与经典环境相比, Savitch 的量性满意度问题( e. 当地汉密尔顿) 的量类比( SCMASPACE) 的量类比( 以 SQSPACE ) 。 为了完整性, 我们引入了SQMAPAC( SQMAPAC ) 的量级比( QARC ) 的量级比( SIMA) 的量级变换算法( ), 将另一个直径解到直径直径解的值的值解的量子值, 直径解到直径解的值, QLUAL QUI 。