12 research outputs found
Two Results about Quantum Messages
We show two results about the relationship between quantum and classical
messages. Our first contribution is to show how to replace a quantum message in
a one-way communication protocol by a deterministic message, establishing that
for all partial Boolean functions we
have . This bound was previously
known for total functions, while for partial functions this improves on results
by Aaronson, in which either a log-factor on the right hand is present, or the
left hand side is , and in which also no entanglement is
allowed.
In our second contribution we investigate the power of quantum proofs over
classical proofs. We give the first example of a scenario, where quantum proofs
lead to exponential savings in computing a Boolean function. The previously
only known separation between the power of quantum and classical proofs is in a
setting where the input is also quantum.
We exhibit a partial Boolean function , such that there is a one-way
quantum communication protocol receiving a quantum proof (i.e., a protocol of
type QMA) that has cost for , whereas every one-way quantum
protocol for receiving a classical proof (protocol of type QCMA) requires
communication
Nearly Optimal Algorithms for Testing and Learning Quantum Junta Channels
We consider the problems of testing and learning quantum -junta channels,
which are -qubit to -qubit quantum channels acting non-trivially on at
most out of qubits and leaving the rest of qubits unchanged. We show
the following.
1. An -query algorithm to distinguish
whether the given channel is -junta channel or is far from any -junta
channels, and a lower bound on the number of
queries;
2. An -query algorithm to learn a -junta
channel, and a lower bound on the number of queries.
This answers an open problem raised by Chen et al. (2023). In order to settle
these problems, we develop a Fourier analysis framework over the space of
superoperators and prove several fundamental properties, which extends the
Fourier analysis over the space of operators introduced in Montanaro and
Osborne (2010)
Simulating Quantum Dynamics On A Quantum Computer
We present efficient quantum algorithms for simulating time-dependent
Hamiltonian evolution of general input states using an oracular model of a
quantum computer. Our algorithms use either constant or adaptively chosen time
steps and are significant because they are the first to have time-complexities
that are comparable to the best known methods for simulating time-independent
Hamiltonian evolution, given appropriate smoothness criteria on the Hamiltonian
are satisfied. We provide a thorough cost analysis of these algorithms that
considers discretizion errors in both the time and the representation of the
Hamiltonian. In addition, we provide the first upper bounds for the error in
Lie-Trotter-Suzuki approximations to unitary evolution operators, that use
adaptively chosen time steps.Comment: Paper modified from previous version to enhance clarity. Comments are
welcom
Query and Depth Upper Bounds for Quantum Unitaries via Grover Search
We prove that any -qubit unitary can be implemented (i) approximately in
time with query access to an appropriate classical
oracle, and also (ii) exactly by a circuit of depth
with one- and two-qubit gates and ancillae. The proofs of (i) and
(ii) involve similar reductions to Grover search. The proof of (ii) also
involves a linear-depth construction of arbitrary quantum states using one- and
two-qubit gates (in fact, this can be improved to constant depth with the
addition of fanout and generalized Toffoli gates) which may be of independent
interest. We also prove a matching lower bound for
(i) and (ii) for a certain class of implementations.Comment: 16 page
On the Hidden Subgroup Problem as a Pivot in Quantum Complexity Theory
Quantum computing has opened the way to new algorithms that can efficiently solve problems that have always been deemed intractable. However, since quantum algorithms are hard to design, the necessity to find a generalization of these problems arises. Such necessity is satisfied by the hidden subgroup problem (HSP), an abstract problem of group theory which successfully generalizes a large number of intractable problems. The HSP plays a significant role in quantum complexity theory, as efficient algorithms that solve it can be employed to efficiently solve other valuable problems, such as integer factorization, discrete logarithms, graph isomorphism and many others.
In this thesis we give a computational definition of the HSP. We then prove the reducibility of some of the aforementioned problems to the HSP. Lastly, we introduce some essential notions of quantum computing and we present two quantum algorithms that efficiently solve the HSP on Abelian groups
Optimal learning of quantum Hamiltonians from high-temperature Gibbs states
We study the problem of learning a Hamiltonian to precision
, supposing we are given copies of its Gibbs state
at a known inverse
temperature . Anshu, Arunachalam, Kuwahara, and Soleimanifar (Nature
Physics, 2021, arXiv:2004.07266) recently studied the sample complexity (number
of copies of needed) of this problem for geometrically local -qubit
Hamiltonians. In the high-temperature (low ) regime, their algorithm has
sample complexity poly and can be implemented with
polynomial, but suboptimal, time complexity.
In this paper, we study the same question for a more general class of
Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error
with sample complexity and
time complexity linear in the sample size, . Furthermore, we prove a
matching lower bound showing that our algorithm's sample complexity is optimal,
and hence our time complexity is also optimal.
In the appendix, we show that virtually the same algorithm can be used to
learn from a real-time evolution unitary in a small regime
with similar sample and time complexity.Comment: 59 pages, v2: incorporated reviewer comments, improved exposition of
appendi
The Complexity of Translationally Invariant Problems Beyond Ground State Energies
The physically motivated quantum generalisation of k-SAT, the k-Local Hamiltonian (k-LH) problem, is well-known to be QMA-complete ("quantum NP"-complete). What is surprising, however, is that while the former is easy on 1D Boolean formulae, the latter remains hard on 1D local Hamiltonians, even if all constraints are identical [Gottesman, Irani, FOCS 2009]. Such "translation-invariant" systems are much closer in structure to what one might see in Nature. Moving beyond k-LH, what is often more physically interesting is the computation of properties of the ground space (i.e. "solution space") itself. In this work, we focus on two such recent problems: Simulating local measurements on the ground space (APX-SIM, analogous to computing properties of optimal solutions to MAX-SAT formulae) [Ambainis, CCC 2014], and deciding if the low energy space has an energy barrier (GSCON, analogous to classical reconfiguration problems) [Gharibian, Sikora, ICALP 2015]. These problems are known to be P^{QMA[log]}- and QCMA-complete, respectively, in the general case. Yet, to date, it is not known whether they remain hard in such simple 1D translationally invariant systems.
In this work, we show that the 1D translationally invariant versions of both APX-SIM and GSCON are intractable, namely are P^{QMA_{EXP}}- and QCMA^{EXP}-complete ("quantum P^{NEXP}" and "quantum NEXP"), respectively. Each of these results is attained by giving a respective generic "lifting theorem". For APX-SIM we give a framework for lifting any abstract local circuit-to-Hamiltonian mapping H satisfying mild assumptions to hardness of APX-SIM on the family of Hamiltonians produced by H, while preserving the structural properties of H (e.g. translation invariance, geometry, locality, etc). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we compress the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we show strong robustness, i.e. soundness even against adversaries acting on all but a single qudit in the system