103 research outputs found
Testing non-isometry is QMA-complete
Determining the worst-case uncertainty added by a quantum circuit is shown to
be computationally intractable. This is the problem of detecting when a quantum
channel implemented as a circuit is close to a linear isometry, and it is shown
to be complete for the complexity class QMA of verifiable quantum computation.
This is done by relating the problem of detecting when a channel is close to an
isometry to the problem of determining how mixed the output of the channel can
be when the input is a pure state. How mixed the output of the channel is can
be detected by a protocol making use of the swap test: this follows from the
fact that an isometry applied twice in parallel does not affect the symmetry of
the input state under the swap operation.Comment: 12 pages, 3 figures. Presentation improved, results unchange
Quantum interactive proofs and the complexity of separability testing
We identify a formal connection between physical problems related to the
detection of separable (unentangled) quantum states and complexity classes in
theoretical computer science. In particular, we show that to nearly every
quantum interactive proof complexity class (including BQP, QMA, QMA(2), and
QSZK), there corresponds a natural separability testing problem that is
complete for that class. Of particular interest is the fact that the problem of
determining whether an isometry can be made to produce a separable state is
either QMA-complete or QMA(2)-complete, depending upon whether the distance
between quantum states is measured by the one-way LOCC norm or the trace norm.
We obtain strong hardness results by proving that for each n-qubit maximally
entangled state there exists a fixed one-way LOCC measurement that
distinguishes it from any separable state with error probability that decays
exponentially in n.Comment: v2: 43 pages, 5 figures, completely rewritten and in Theory of
Computing (ToC) journal forma
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
Constant-Soundness Interactive Proofs for Local Hamiltonians
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 is
replaced by a pair of functions \Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C),
the set of -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
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Universal Quantum Hamiltonians
Quantum many-body systems exhibit an extremely diverse range of phases and
physical phenomena. Here, we prove that the entire physics of any other quantum
many-body system is replicated in certain simple, "universal" spin-lattice
models. We first characterise precisely what it means for one quantum many-body
system to replicate the entire physics of another. We then show that certain
very simple spin-lattice models are universal in this very strong sense.
Examples include the Heisenberg and XY models on a 2D square lattice (with
non-uniform coupling strengths). We go on to fully classify all two-qubit
interactions, determining which are universal and which can only simulate more
restricted classes of models. Our results put the practical field of analogue
Hamiltonian simulation on a rigorous footing and take a significant step
towards justifying why error correction may not be required for this
application of quantum information technology.Comment: 78 pages, 9 figures, 44 theorems etc. v2: Trivial fixes. v3: updated
and simplified proof of Thm. 9; 82 pages, 47 theorems etc. v3: Small fix in
proof of time-evolution lemma (this fix not in published version
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