Quantum space, ground space traversal, and how to embed multi-prover interactive proofs into unentanglement
S. Gharibian, D. Rudolph, ArXiv:2206.05243 (2022).
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Gharibian, SevagLibreCat
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Rudolph, Dorian

Abstract
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.
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Journal Title
arXiv:2206.05243
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Cite this
Gharibian S, Rudolph D. Quantum space, ground space traversal, and how to embed multi-prover interactive proofs into unentanglement. arXiv:220605243. Published online 2022.
Gharibian, S., & Rudolph, D. (2022). Quantum space, ground space traversal, and how to embed multi-prover interactive proofs into unentanglement. In arXiv:2206.05243.
@article{Gharibian_Rudolph_2022, title={Quantum space, ground space traversal, and how to embed multi-prover interactive proofs into unentanglement}, journal={arXiv:2206.05243}, author={Gharibian, Sevag and Rudolph, Dorian}, year={2022} }
Gharibian, Sevag, and Dorian Rudolph. “Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement.” ArXiv:2206.05243, 2022.
S. Gharibian and D. Rudolph, “Quantum space, ground space traversal, and how to embed multi-prover interactive proofs into unentanglement,” arXiv:2206.05243. 2022.
Gharibian, Sevag, and Dorian Rudolph. “Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement.” ArXiv:2206.05243, 2022.