672 research outputs found
The Quantum Complexity of Computing Schatten p-norms
We consider the quantum complexity of computing Schatten -norms and
related quantities, and find that the problem of estimating these quantities is
closely related to the one clean qubit model of computation. We show that the
problem of approximating for a log-local -qubit
Hamiltonian and , up to a suitable level of accuracy, is
contained in DQC1; and that approximating this quantity up to a somewhat higher
level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by
the quantum algorithm is substantially better than a natural classical
algorithm for the problem. The same problem can be solved for arbitrary sparse
matrices in BQP. One application of the algorithm is the approximate
computation of the energy of a graph.Comment: 28 page
Estimating operator norms using covering nets
We present several polynomial- and quasipolynomial-time approximation schemes
for a large class of generalized operator norms. Special cases include the
norm of matrices for , the support function of the set of
separable quantum states, finding the least noisy output of
entanglement-breaking quantum channels, and approximating the injective tensor
norm for a map between two Banach spaces whose factorization norm through
is bounded.
These reproduce and in some cases improve upon the performance of previous
algorithms by Brand\~ao-Christandl-Yard and followup work, which were based on
the Sum-of-Squares hierarchy and whose analysis used techniques from quantum
information such as the monogamy principle of entanglement. Our algorithms, by
contrast, are based on brute force enumeration over carefully chosen covering
nets. These have the advantage of using less memory, having much simpler proofs
and giving new geometric insights into the problem. Net-based algorithms for
similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in
each case with a run-time that is exponential in the rank of some matrix. We
achieve polynomial or quasipolynomial runtimes by using the much smaller nets
that exist in spaces. This principle has been used in learning theory,
where it is known as Maurey's empirical method.Comment: 24 page
Quantum Approximation of Normalized Schatten Norms and Applications to Learning
Efficient measures to determine similarity of quantum states, such as the
fidelity metric, have been widely studied. In this paper, we address the
problem of defining a similarity measure for quantum operations that can be
\textit{efficiently estimated}. Given two quantum operations, and ,
represented in their circuit forms, we first develop a quantum sampling circuit
to estimate the normalized Schatten 2-norm of their difference () with precision , using only one clean qubit and one
classical random variable. We prove a Poly upper bound on
the sample complexity, which is independent of the size of the quantum system.
We then show that such a similarity metric is directly related to a functional
definition of similarity of unitary operations using the conventional fidelity
metric of quantum states (): If is sufficiently small
(e.g. ) then the fidelity of
states obtained by processing the same randomly and uniformly picked pure
state, , is as high as needed () with probability exceeding . We
provide example applications of this efficient similarity metric estimation
framework to quantum circuit learning tasks, such as finding the square root of
a given unitary operation.Comment: 25 pages, 4 figures, 6 tables, 1 algorith
Circuit Complexity and 2D Bosonisation
We consider the circuit complexity of free bosons, or equivalently free
fermions, in 1+1 dimensions. Motivated by the results of [1] and [2, 3] who
found different behavior in the complexity of free bosons and fermions, in any
dimension, we consider the 1+1 dimensional case where, thanks to the
bosonisation equivalence, we can consider the same state from both the bosonic
and the fermionic perspectives. In this way the discrepancy can be attributed
to a different choice of the set of gates allowed in the circuit. We study the
effect in two classes of states: i) bosonic-coherent / fermionic-gaussian
states; ii) states that are both bosonic- and fermionic-gaussian. We consider
the complexity relative to the ground state. In the first class, the different
results can be reconciled admitting a mode-dependent cost function in one of
the descriptions. The differences in the second class are more important, in
terms of the cutoff-dependence and the overall behavior of the complexity.Comment: Fix typos and add reference
A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs
The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier
analysis of real-valued functions on the Boolean cube. In this paper we present
a version of this inequality for matrix-valued functions on the Boolean cube.
Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also
present a number of applications. First, we analyze maps that encode
classical bits into qubits, in such a way that each set of bits can be
recovered with some probability by an appropriate measurement on the quantum
encoding; we show that if , then the success probability is
exponentially small in . This result may be viewed as a direct product
version of Nayak's quantum random access code bound. It in turn implies strong
direct product theorems for the one-way quantum communication complexity of
Disjointness and other problems. Second, we prove that error-correcting codes
that are locally decodable with 2 queries require length exponential in the
length of the encoded string. This gives what is arguably the first
``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf
using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the
proceedings of the IEEE FOCS 08 conferenc
Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement
We initiate the study of quantifying nonlocalness of a bipartite measurement
by the minimum amount of classical communication required to simulate the
measurement. We derive general upper bounds, which are expressed in terms of
certain tensor norms of the measurement operator. As applications, we show that
(a) If the amount of communication is constant, quantum and classical
communication protocols with unlimited amount of shared entanglement or shared
randomness compute the same set of functions; (b) A local hidden variable model
needs only a constant amount of communication to create, within an arbitrarily
small statistical distance, a distribution resulted from local measurements of
an entangled quantum state, as long as the number of measurement outcomes is
constant.Comment: A preliminary version of this paper appears as part of an article in
Proceedings of the the 37th ACM Symposium on Theory of Computing (STOC 2005),
460--467, 200
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