Constant-Soundness Interactive Proofs for Local Hamiltonians

$\newcommand{\Xlin}{\mathcal{X}} \newcommand{\Zlin}{\mathcal{Z}} \newcommand{\C}{\mathbb{C}}$We give a quantum multiprover interactive proof system for the local Hamiltonian problem in which there is a constant number of provers, questions are classical of length polynomial in the number of qubits, and answers are of constant length. The main novelty of our protocol is that the gap between completeness and soundness is directly proportional to the promise gap on the (normalized) ground state energy of the Hamiltonian. This result can be interpreted as a concrete step towards a quantum PCP theorem giving entangled-prover interactive proof systems for QMA-complete problems. The key ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld, where the function $f:\{0,1\}^n\to\{0,1\}$ is replaced by a pair of functions \Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C), the set of $d$-dimensional Hermitian matrices that square to identity. The test enforces that (i) each function is exactly linear, \Xlin(a)\Xlin(b)=\Xlin(a+b) and \Zlin(a) \Zlin(b)=\Zlin(a+b), and (ii) the two functions are approximately complementary, \Xlin(a)\Zlin(b)\approx (-1)^{a\cdot b} \Zlin(b)\Xlin(a).Comment: 33 page

The Quantum PCP Conjecture

The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics, the global nature of entanglement and its topological properties, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column from Volume 44 Issue 2, June 201

Robust self-testing of many-qubit states

We introduce a simple two-player test which certifies that the players apply tensor products of Pauli $\sigma_X$ and $\sigma_Z$ observables on the tensor product of $n$ EPR pairs. The test has constant robustness: any strategy achieving success probability within an additive $\varepsilon$ of the optimal must be $\mathrm{poly}(\varepsilon)$-close, in the appropriate distance measure, to the honest $n$-qubit strategy. The test involves $2n$-bit questions and $2$-bit answers. The key technical ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld. As applications of our result we give (i) the first robust self-test for $n$ EPR pairs; (ii) a quantum multiprover interactive proof system for the local Hamiltonian problem with a constant number of provers and classical questions and answers, and a constant completeness-soundness gap independent of system size; (iii) a robust protocol for delegated quantum computation.Comment: 36 pages. Improves upon and supersedes our earlier submission arXiv:1512.0209

Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement

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.Comment: 60 pages, 4 figure