18 research outputs found
A Converse for Fault-tolerant Quantum Computation
With improvements in achievable redundancy for fault-tolerant quantum
computing, it is natural to ask: what is the minimum required redundancy? In
this paper, we obtain a lower bound on the minimum redundancy required for
-accurate implementation of a large class of operations, which
includes unitary operators. For the practically relevant case of
sub-exponential (in input size) depth and sub-linear gate size, our bound on
redundancy is tighter than the best known lower bound in \cite{FawziMS2022}. We
obtain this bound by connecting fault-tolerant computation with a set of finite
blocklength quantum communication problems whose accuracy requirements satisfy
a joint constraint. This bound gives a strictly lower noise threshold for
non-degradable noise and captures its dependence on gate size. This bound
directly extends to the case where noise at the outputs of a gate are
correlated but noise across gates are independent.Comment: 10 page
The Role of Correlated Noise in Quantum Computing
This paper aims to give an overview of the current state of fault-tolerant
quantum computing, by surveying a number of results in the field. We show that
thresholds can be obtained for a simple noise model as first proved in [AB97,
Kit97, KLZ98], by presenting a proof for statistically independent noise,
following the presentation of Aliferis, Gottesman and Preskill [AGP06]. We also
present a result by Terhal and Burkard [TB05] and later improved upon by
Aliferis, Gottesman and Preskill [AGP06] that shows a threshold can still be
obtained for local non-Markovian noise, where we allow the noise to be weakly
correlated in space and time. We then turn to negative results, presenting work
by Ben-Aroya and Ta-Shma [BT11] who showed conditional errors cannot be
perfectly corrected. We end our survey by briefly mentioning some more
speculative objections, as put forth by Kalai [Kal08, Kal09, Kal11]
Achieving quantum supremacy with sparse and noisy commuting quantum computations
The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome
Quantum boolean functions
In this paper we introduce the study of quantum boolean functions, which are
unitary operators f whose square is the identity: f^2 = I. We describe several
generalisations of well-known results in the theory of boolean functions,
including quantum property testing; a quantum version of the Goldreich-Levin
algorithm for finding the large Fourier coefficients of boolean functions; and
two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier
spectra of boolean functions. In order to obtain one of these generalisations,
we prove a quantum extension of the hypercontractive inequality of Bonami,
Gross and Beckner.Comment: 47 pages; v5: fixes previously corrupt fil