30 research outputs found
Approximation, Proof Systems, and Correlations in a Quantum World
This thesis studies three topics in quantum computation and information: The
approximability of quantum problems, quantum proof systems, and non-classical
correlations in quantum systems.
In the first area, we demonstrate a polynomial-time (classical) approximation
algorithm for dense instances of the canonical QMA-complete quantum constraint
satisfaction problem, the local Hamiltonian problem. In the opposite direction,
we next introduce a quantum generalization of the polynomial-time hierarchy,
and define problems which we prove are not only complete for the second level
of this hierarchy, but are in fact hard to approximate.
In the second area, we study variants of the interesting and stubbornly open
question of whether a quantum proof system with multiple unentangled quantum
provers is equal in expressive power to a proof system with a single quantum
prover. Our results concern classes such as BellQMA(poly), and include a novel
proof of perfect parallel repetition for SepQMA(m) based on cone programming
duality.
In the third area, we study non-classical quantum correlations beyond
entanglement, often dubbed "non-classicality". Among our results are two novel
schemes for quantifying non-classicality: The first proposes the new paradigm
of exploiting local unitary operations to study non-classical correlations, and
the second introduces a protocol through which non-classical correlations in a
starting system can be "activated" into distillable entanglement with an
ancilla system.
An introduction to all required linear algebra and quantum mechanics is
included.Comment: PhD Thesis, 240 page
Strong NP-Hardness of the Quantum Separability Problem
Given the density matrix rho of a bipartite quantum state, the quantum
separability problem asks whether rho is entangled or separable. In 2003,
Gurvits showed that this problem is NP-hard if rho is located within an inverse
exponential (with respect to dimension) distance from the border of the set of
separable quantum states. In this paper, we extend this NP-hardness to an
inverse polynomial distance from the separable set. The result follows from a
simple combination of works by Gurvits, Ioannou, and Liu. We apply our result
to show (1) an immediate lower bound on the maximum distance between a bound
entangled state and the separable set (assuming P != NP), and (2) NP-hardness
for the problem of determining whether a completely positive trace-preserving
linear map is entanglement-breaking.Comment: 18 pages, 1 figure. v5: Updated version to appear in Quantum
Information & Computation. Includes additional details in proof of
NP-hardness of determining whether a quantum channel is
entanglement-breaking, as well as minor updates to improve readability
throughout. Thank you to anonymous referees for their comment
Almost Optimal Classical Approximation Algorithms for a Quantum Generalization of Max-Cut
Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-feromagnetic XY model) and 0.498 (anti-ferromagnetic Heisenberg XYZ model). These are almost optimal; we show that the best possible ratios achievable by a product state for these models is 2/3 and 1/2, respectively
Oracle Complexity Classes and Local Measurements on Physical Hamiltonians
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]-complete even when restricted
to more physical Hamiltonians, obtaining as intermediate steps a variety of
related complexity-theoretic results.
We first give a sequence of results which together yield P^QMA[log]-hardness
for APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA,
and QMA oracles, a logarithmic number of adaptive queries is equivalent to
polynomially many parallel queries. These equalities simplify the proofs of our
subsequent results. (2) Next, we show that the hardness of APX-SIM is preserved
under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a
byproduct, we obtain a full complexity classification of APX-SIM, showing it is
complete for P, P^||NP, P^||StoqMA, or P^||QMA depending on the Hamiltonians
employed. (3) Leveraging the above, we show that APX-SIM is P^QMA[log]-complete
for any family of Hamiltonians which can efficiently simulate spatially sparse
Hamiltonians, including physically motivated models such as the 2D Heisenberg
model.
Our second focus considers 1D systems: We show that APX-SIM remains
P^QMA[log]-complete even for local Hamiltonians on a 1D line of 8-dimensional
qudits. This uses a number of ideas from above, along with replacing the "query
Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.Comment: 38 pages, 3 figure
On Polynomially Many Queries to NP or QMA Oracles
We study the complexity of problems solvable in deterministic polynomial time
with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as
and , respectively. The former allows one to classify problems more
finely than the Polynomial-Time Hierarchy (PH), whereas the latter
characterizes physically motivated problems such as Approximate Simulation
(APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by
the classes and , defined identically to
and , except that only logarithmically many oracle queries are
allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by
a machine have a "query graph" which is a tree, then this computation
can be simulated in .
In this work, we first show that for any verification class
, any machine with a query
graph of "separator number" can be simulated using deterministic time
and queries to a -oracle. When (which
includes the case of -treewidth, and thus also of trees), this gives an
upper bound of , and when , this yields bound
(QP meaning quasi-polynomial time). We next show how to
combine Gottlob's "admissible-weighting function" framework with the
"flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a
unified approach for embedding computations directly into APX-SIM
instances in a black-box fashion. Finally, we formalize a simple no-go
statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear
polynomial specified via an arithmetic circuit, if one can "weakly
compress" so that its optimal value requires bits to represent, then
can be decided with only queries to an NP-oracle.Comment: 46 pages pages, 5 figures, to appear in ITCS 202
The Complexity of Simulating Local Measurements on Quantum Systems
An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, Ambainis 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 results.
The P^QMA[log]-completeness result of Ambainis requires O(log n)-local observ- ables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P^QMA[log]-complete, resolving an open question of Ambainis. We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P^QMA[log]-complete.
P^QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of Beigel, Hemachandra, and Wechsung to show P^QMA[log] subseteq PP. This improves the containment QMA subseteq PP from Kitaev and Watrous. A central theme of this work is the subtlety involved in the study of oracle classes in which the oracle solves a promise problem. In this vein, we identify a flaw in Ambainis\u27 prior work regarding a P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on his prior work to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions
Gate-efficient discrete simulations of continuous-time quantum query algorithms
We show how to efficiently simulate continuous-time quantum query algorithms
that run in time T in a manner that preserves the query complexity (within a
polylogarithmic factor) while also incurring a small overhead cost in the total
number of gates between queries. By small overhead, we mean T within a factor
that is polylogarithmic in terms of T and a cost measure that reflects the cost
of computing the driving Hamiltonian. This permits any continuous-time quantum
algorithm based on an efficiently computable driving Hamiltonian to be
converted into a gate-efficient algorithm with similar running time.Comment: 28 pages, 2 figure
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