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### Simulating quantum field theories on continuous-variable quantum computers

##### Steven Abel, Michael Spannowsky, and Simon Williams

##### Phys. Rev. A **110**, 012607 – Published 8 July 2024

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#### Abstract

We delve into the use of photonic quantum computing to simulate quantum mechanics and extend its application towards quantum field theory. We develop and prove a method that leverages this form of continuous-variable quantum computing (CVQC) to reproduce the time evolution of quantum-mechanical states under arbitrary Hamiltonians, and we demonstrate the method's remarkable efficacy with various potentials. Our method centers on constructing an *evolver state*, a specially prepared quantum state that induces the desired time evolution on the target state. This is achieved by introducing a non-Gaussian operation using a measurement-based quantum computing approach, enhanced by machine learning. Furthermore, we propose a framework in which these methods can be extended to encode field theories in CVQC without discretizing the field values, thus preserving the continuous nature of the fields. This opens new avenues for quantum computing applications in quantum field theory.

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- Received 29 March 2024
- Accepted 18 June 2024

DOI:https://doi.org/10.1103/PhysRevA.110.012607

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

#### Physics Subject Headings (PhySH)

- Research Areas

Quantum circuitsQuantum field theoryQuantum simulation

Particles & FieldsQuantum Information, Science & Technology

#### Authors & Affiliations

Steven Abel^{1,2,*}, Michael Spannowsky^{1,†}, and Simon Williams^{1,‡}

^{1}Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, England, United Kingdom^{2}Department of Mathematical Sciences, Durham University, Durham DH1 3LE, England, United Kingdom

^{*}Contact author: steve.abel@durham.ac.uk^{†}Contact author: michael.spannowsky@durham.ac.uk^{‡}Contact author: simon.j.williams@durham.ac.uk

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#### Images

###### Figure 1

Example operations using a squeezing gate, and a controlled-$X$ gate together with hom*odyne measurement. In (a)the ground state is squeezed by $S[ln(1/2\left)\right]$ which produces a wave function flattened by a factor of 2 (yellow line). Then it is squeezed again by $S[ln\phantom{\rule{0ex}{0ex}}(4\left)\right]$ producing a ground state squeezed by a factor 2 (orange line). In (b)we perform a composite displacement, by using a controlled-$X$ gate followed by a hom*odyne measurement of $y$, first displacing by 2 to the right, so that $\psi \left(x\right)\to \psi (x-2)$ (yellow line), then displacing by 4 to the left, so that $\psi (x-2)\to \psi (x+4)$ (red line). In this case the Fock truncation is 60, and we see some distortion beginning to appear at the peak.

###### Figure 2

Circuit diagram representation of the simple manipulations which produced Fig.1. Following the convention for control gates in discrete gate systems, the controlled-$X$ gate in (b)is controlled by the upper $y$ qumode (represented by a solid circle) and acts on the lower $x$ qumode (represented by crosshairs).

###### Figure 3

The potentials of focus in this study, namely, the (a)quartic, (b)$cosh$ potential, and (c)$w$ potential. In all cases the wave function is initialized as a Gaussian state centered on the origin.

###### Figure 4

Evolver gadget to evolve through a single Trotter step. Here $|\varphi \rangle $ is the evolver state which is set according to Eq.(3.19), and which in Sec.3b will be machine learned using a measurement-based quantum algorithm.

###### Figure 5

Schematic of a quantum circuit for the preparation of a non-Gaussian state. The circuit architecture is inspired by a Gaussian boson sampling routine on $N$ qumodes. The incoming vacuum states are displaced, then squeezed before being interfaced with an interferometer, constructed using the rectangular architecture from Ref.[40]. This routine is repeated for $I$ layers, parametrized with trainable variables, ${\theta}_{i}$, for each layer. Finally, ($n-1$)-measurements are made using PNR detectors, with the $j\mathrm{th}$ measurement postselecting on ${m}_{j}$. These measurements generate the induced non-Gaussian state on the $n$th qumode.

###### Figure 6

Schematic of a three-qumode quantum circuit for the preparation of the non-Gaussian evolver state.

###### Figure 7

Normalized evolver state function with a truncation to the first 25 Fock levels. The state has been trained using an $N$-qumode Gaussian boson sampling architecture with $(N-1)$-measurements, schematically shown in Fig.5, to produce the evolver state on the $N$th qumode.

###### Figure 8

The time evolution of a quantum system with the asymmetric quartic potential of Eq.(3.3) with (a)$\varepsilon =0.1$ and (b)$\varepsilon =0.5$, generated by the photonic quantum simulator with a Fock truncation of 60 (solid line) and compared to an exact calculation (dotted lines).

###### Figure 9

The KL divergence between the quantum simulation and exact calculation for different evolution times and Fock truncations for the (a)quartic, (b)$cosh$, and (c)$w$ potentials. The KL divergence quantifies the disparity between probability distributions as a relative entropy (which broadly speaking encodes the information required to get from one distribution to the other). After a sufficient cutoff, the KL divergence exhibits a monotonic behavior with time.

###### Figure 10

The time evolution of a quantum system under the influence of the hyperbolic potential from Eq.(3.4) generated by the photonic quantum simulator (solid lines) compared to an exact calculation (dotted lines). In (a)the circuit has used the Ket command to initialize the evolver state and has been run at a Fock truncation of 60. In (b)the evolver state has been initialized using the full circuit and has been run at a truncation of 25.

###### Figure 11

The time evolution of a quantum system under the influence of the $w$ potential of Eq.(3.5) generated by the photonic quantum simulator (solid lines) compared to an exact calculation (dotted lines).

###### Figure 12

A comparison between the time evolution simulated using the full circuit (solid lines) and the Ket command (dotted lines) for a Fock truncation of ${n}_{\mathrm{max}}=25$.