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Register a new userSubspace projection method for unstructured searches with noisy quantum oracles using a signal-based quantum emulation device
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This paper describes a novel approach to solving unstructured search problems using a classical, signal-based emulation of a quantum computer. The classical nature of the representation allows one to perform subspace projections in addition to the usual unitary gate operations. Although bandwidth requirements will limit the scale of problems that can be solved by this method, it can nevertheless provide a significant computational advantage for problems of limited size. In particular, we find that, for the same number of noisy oracle calls, the proposed subspace projection method provides a higher probability of success for finding a solution than does an single application of Grovers algorithm on the same device.
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We study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using $tilde{O}(1)$ quantum queries to a membership oracle, which is an exponential quantum speed-up over the $Omega(n)$ membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with a simplification of recent classical work of Lee, Sidford, and Vempala gives our efficient separation oracle. This in turn implies, via a known algorithm, that $tilde{O}(n)$ quantum queries to a membership oracle suffice to implement an optimization oracle (the best known classical upper bound on the number of membership queries is quadratic). We also prove several lower bounds: $Omega(sqrt{n})$ quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and $Omega(n)$ quantum separation queries are needed if it does not.
A standard quantum oracle $S_f$ for a general function $f: Z_N to Z_N $ is defined to act on two input states and return two outputs, with inputs $ket{i}$ and $ket{j}$ ($i,j in Z_N $) returning outputs $ket{i}$ and $ket{j oplus f(i)}$. However, if $f$ is known to be a one-to-one function, a simpler oracle, $M_f$, which returns $ket{f(i)}$ given $ket{i}$, can also be defined. We consider the relative strengths of these oracles. We define a simple promise problem which minimal quantum oracles can solve exponentially faster than classical oracles, via an algorithm which cannot be naively adapted to standard quantum oracles. We show that $S_f$ can be constructed by invoking $M_f$ and $(M_f)^{-1}$ once each, while $Theta(sqrt{N})$ invocations of $S_f$ and/or $(S_f)^{-1}$ are required to construct $M_f$.
We consider the hypothesis that quantum mechanics is not fundamental, but instead emerges from a theory with less computational power, such as classical mechanics. This hypothesis makes the prediction that quantum computers will not be capable of sufficiently complex quantum computations. Utilizing this prediction, we outline a proposal to test for such a breakdown of quantum mechanics using near-term noisy intermediate-scale quantum (NISQ) computers. Our procedure involves simulating a non-Clifford random circuit, followed by its inverse, and then checking that the resulting state is the same as the initial state. We show that quantum mechanics predicts that the fidelity of this procedure decays exponentially with circuit depth (due to noise in NISQ computers). However, if quantum mechanics emerges from a theory with significantly less computational power, then we expect the fidelity to decay significantly more rapidly than the quantum mechanics prediction for sufficiently deep circuits, which is the experimental signature that we propose to search for. Useful experiments can be performed with 80 qubits and gate infidelity $10^{-3}$, while highly informative experiments should require only 1000 qubits and gate infidelity $10^{-5}$.
This paper describes a novel approach to emulate a universal quantum computer with a wholly classical system, one that uses a signal of bounded duration and amplitude to represent an arbitrary quantum state. The signal may be of any modality (e.g. acoustic, electromagnetic, etc.) but this paper will focus on electronic signals. Individual qubits are represented by in-phase and quadrature sinusoidal signals, while unitary gate operations are performed using simple analog electronic circuit devices. In this manner, the Hilbert space structure of a multi-qubit quantum state, as well as a universal set of gate operations, may be fully emulated classically. Results from a programmable prototype system are presented and discussed.
Quantum computers provide a fundamentally new computing paradigm that promises to revolutionize our ability to solve broad classes of problems. Surprisingly, the basic mathematical structures of gate-based quantum computing, such as unitary operations on a finite-dimensional Hilbert space, are not unique to quantum systems but may be found in certain classical systems as well. Previously, it has been shown that one can represent an arbitrary multi-qubit quantum state in terms of classical analog signals using nested quadrature amplitude modulated signals. Furthermore, using digitally controlled analog electronics one may manipulate these signals to perform quantum gate operations and thereby execute quantum algorithms. The computational capacity of a single signal is, however, limited by the required bandwidth, which scales exponentially with the number of qubits when represented using frequency-based encoding. To overcome this limitation, we introduce a method to extend this approach to multiple parallel signals. Doing so allows a larger quantum state to be emulated with the same gate time required for processing frequency-encoded signals. In the proposed representation, each doubling of the number of signals corresponds to an additional qubit in the spatial domain. Single quit gate operations are similarly extended so as to operate on qubits represented using either frequency-based or spatial encoding schemes. Furthermore, we describe a method to perform gate operations between pairs of qubits represented using frequency or spatial encoding or between frequency-based and spatially encoded qubits. Finally, we describe how this approach may be extended to represent qubits in the time domain as well.
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