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Journal of Physics A Mathematical and General

Construction of quantum states from an optimally truncated von Neumann lattice of coherent states

Lampros K. Stergioulas, Vassilios S. Vassiliadis, Apostol Vourdas

Published Jan 1, 1999• 7 citations
Publisher

Abstract

Arbitrary quantum states can be expanded with a very good accuracy in terms of coherent states on a truncated von Neumann lattice. Optimization techniques are used to further reduce the size of the basis, whilst still preserving the same degree of accuracy. Various p -norms are adopted as accuracy measures, which reflect different practical needs. Numerical examples demonstrate that a significant reduction in the size of the basis can be achieved with optimization techniques, without substantial loss of accuracy.

DOI
10.1088/0305-4470/32/17/308

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Arbitrary quantum states can be expanded with a very good accuracy in terms of coherent states on a truncated von Neumann lattice. Optimization techniques are used to further reduce the size of the basis, whilst still preserving the same degree of accuracy. Various p -norms are adopted as accuracy measures, which reflect different practical needs. Numerical examples demonstrate that a significant reduction in the size of the basis can be achieved with optimization techniques, without substantial loss of accuracy.
Measurements on near-term quantum processors are inevitably subject to hardware imperfections that lead to readout errors. Mitigation of such unavoidable errors is crucial to better explore and extend the power of near-term quantum hardware. In this work, we propose a method to mitigate measurement errors in computing quantum expectation values using the truncated Neumann series. The essential idea is to cancel the errors by combining various noisy expectation values generated by sequential measurements determined by terms in the truncated series. We numerically test this method and find that the computation accuracy is substantially improved. Our method possesses several advantages: it does not assume any noise structure, it does not require the calibration procedure to learn the noise matrix a prior, and most importantly, the incurred error mitigation overhead is independent of system size, as long as the noise resistance of the measurement device is moderate. All these advantages empower our method as a practical measurement error mitigation method for near-term quantum devices.
An orthonormal basis of modified coherent states is constructed. Each member of the basis is an infinite sum of coherent states on a von Neumann lattice. A single state is assigned to each unit cell of area h (Planck constant) in the phase plane. The uncertainties of the coordinate x and the square of the momentum p 2 for these states are shown to be similar to those for the usual coherent states. Expansions in the newly established set are discussed and it is shown that any function in the kq -representation can be written as a sum of two fixed kq -functions. Approximate commuting operators for x and p 2 are defined on a lattice in phase plane according to von Neumann's prescription.
Adaptive agents realise complex reactive behaviours by using a memory of past input stimuli and output actions to guide structured future responses. Quantum adaptive agents can operate while storing less information in memory than optimal classical counterparts; yet, this does not necessarily translate into a reduced dimension of the memory that must be physically realised. We introduce a route-truncate-repair procedure that can convert entropic quantum memory advantages into reductions in memory dimension. Routing a reference input process through an agent yields a temporal matrix product state representation whose canonical bond is identified with the joint reference--agent memory. Truncating the agent's share of this bond and locally repairing the resulting dynamics produces a smaller, physically-valid agent that remains capable of responding to arbitrary input sequences. A fidelity-divergence certificate quantifies the resulting trade-off between accuracy and memory dimension. Benchmark adaptive processes exhibit substantial dimension reduction whilst preserving the underlying behaviour with high fidelity. The construction extends to feedback and coherent quantum interactions, with accuracy guarantees under arbitrary adaptive interrogation for a class of agents. For a resettable clock, four quantum memory dimensions attain an input--output fidelity divergence rate first matched with eleven states in our search over classical reduced models. These results establish a route from entropic memory advantages to practical, dimension-reduced adaptive quantum agents.