Abstract
Purpose
This study aims to assess the use of variational quantum imaginary time evolution for solving partial differential equations using real-amplitude ansätze with full circular entangling layers. A graphical mapping technique for encoding impulse functions is also proposed.
Design/methodology/approach
The Smoluchowski equation, including the Derjaguin–Landau–Verwey–Overbeek potential energy, is solved to simulate colloidal deposition on a planar wall. The performance of different types of entangling layers and over-parameterization is evaluated.
Findings
Colloidal transport can be modelled adequately with variational quantum simulations. Full circular entangling layers with real-amplitude ansätze lead to higher-fidelity solutions. In most cases, the proposed graphical mapping technique requires only a single bit-flip with a parametric gate. Over-parameterization is necessary to satisfy certain physical boundary conditions, and higher-order time-stepping reduces norm errors.
Practical implications
Variational quantum simulation can solve partial differential equations using near-term quantum devices. The proposed graphical mapping technique could potentially aid quantum simulations for certain applications.
Originality/value
This study shows a concrete application of variational quantum simulation methods in solving practically relevant partial differential equations. It also provides insight into the performance of different types of entangling layers and over-parameterization. The proposed graphical mapping technique could be valuable for quantum simulation implementations. The findings contribute to the growing body of research on using variational quantum simulations for solving partial differential equations.
Keywords
Citation
Leong, F.Y., Koh, D.E., Ewe, W.-B. and Kong, J.F. (2023), "Variational quantum simulation of partial differential equations: applications in colloidal transport", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 33 No. 11, pp. 3669-3690. https://doi.org/10.1108/HFF-05-2023-0265
Publisher
:Emerald Publishing Limited
Copyright © 2023, Fong Yew Leong, Dax Enshan Koh, Wei-Bin Ewe and Jian Feng Kong.
License
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial & non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Partial differential equations (PDEs) are fundamental to solving important problems in engineering and science. With the advent of nascent quantum computers, finding new efficient quantum algorithms and hardware for solving PDEs has become an active area of research (Tosti Balducci et al., 2022; Jin et al., 2022; Leong et al., 2022; Pool et al., 2022) in disciplines ranging from fluid dynamics (Budinski, 2021; Gaitan, 2020; Steijl, 2022; Steijl and Barakos, 2018; Griffin et al., 2019; Li et al., 2023), heat conduction (Liu et al., 2022) and electromagnetics (Ewe et al., 2022) to quantitative finance (Fontanela et al., 2021) and cosmology (Mocz and Szasz, 2021).
Although linear differential equations can be solved by the quantum linear solver algorithm (Berry et al., 2017; Harrow et al., 2009), the required resources are out of reach of the current noisy intermediate-scale quantum devices (Lau et al., 2022; Bharti et al., 2022; Preskill, 2018). In fact, practical near-term quantum algorithms are limited to those designed for short circuit depths, such as variational quantum algorithms (VQAs) (Cerezo et al., 2021), which use parameterized ansätze to optimize cost functions via variational updating.
VQAs can largely be classified into two categories, namely, optimization and simulation (Endo et al., 2021), each offering unique approaches to solving PDEs. Variational quantum optimization (VQO) aims to optimize a static target cost function through parameter tuning, an example of which is the popular variational quantum eigensolver (VQE) (Peruzzo et al., 2014) for minimizing energy states in the field of quantum chemistry. This led to the development of the variational quantum linear equation solver (Bravo-Prieto et al., 2019; Huang et al., 2021; Xu et al., 2021) for systems of linear equations and the variational quantum Poisson solver (Liu et al., 2021; Sato et al., 2021). Evolution of the Poisson equation allows parabolic PDEs to be solved through implicit time-stepping (Leong et al., 2022), which requires quantum information to be updated and encoded at each time-step.
On the other hand, variational quantum simulation (VQS) aims to simulate a dynamical quantum process, such as the Schrödinger time evolution (Li and Benjamin, 2017). This allows certain PDEs to be solved efficiently using imaginary quantum time evolution (Endo et al., 2021; Endo et al., 2020; Yuan et al., 2019), including the Black–Scholes equation for option pricing (Miyamoto and Kubo, 2021; Radha, 2021; Stamatopoulos et al., 2020) and stochastic differential equations for stochastic processes (Kubo et al., 2021). Recent work on the Feynman–Kac formulation (Alghassi et al., 2022) generalizes quantum simulation of parabolic PDEs, paving the way for potential near-term applications.
In this study, we explore applications of VQS (Miyamoto and Kubo, 2021; Alghassi et al., 2022) in solving PDEs, including the Smoluchowski equation for colloidal physics, with an emphasis on potential and non-homogeneous terms oft-neglected in quantum simulations. We select for high-fidelity real-amplitude ansätze, assess time complexity and propose an efficient encoding scheme for idealized pulse functions, as a proof of concept towards practical implementation of quantum simulation.
2. Variational quantum simulation
2.1 Evolution equation
Consider a 1-dimensional (1D) evolution equation expressed in the Feynman–Kac formulation (Alghassi et al., 2022):
With A and C specified, parameters θ are evolved in time using the forward Euler method as:
2.2 Decomposition of Hamiltonian
The Hamiltonian operator
The operator S denotes the n-qubit cyclic shift operator (Sato et al., 2021):
2.3 Ansatz selection
For optimal algorithmic performance, a good choice of ansatz is crucial (Tilly et al., 2022; You et al., 2021). For PDEs that admit only real solutions, it is preferable to use a real-amplitude ansatz formed by nl repeating blocks, each one consisting of a parameterized layer with one RY rotation gate on each qubit, followed by an entangling layer with CNOT gates between consecutive qubits (Alghassi et al., 2022). Here, we consider two options for customization: the first between linear and circular entanglement and the second with or without an unentangled parameterized layer as the final block nl, as shown in Figure 1.
We note that the circular entanglement with a final unentangled layer [Figure 1(c)] is a popular choice of an ansatz for VQS (Kubo et al., 2021; Alghassi et al., 2022). For benchmarking, we perform numerical experiments on the various ansätze (Figure 1) to solve a simple 1D heat or diffusion equation, expressed in Dirac notation as:
The initial trial state is set as a reverse step function (Sato et al., 2021):
We measure the fidelity of the VQS solution obtained from each ansatz compared to the classical solution and define the trace error as:
Similarly, we define the norm error as:
The circular, fully entangled ansatz [Figure 1(d)], here termed full circular ansatz for short, was found to outperform other ansätze, requiring fewer parameters for the same solution fidelity. For four qubits, the full circular ansatz is the only one able to produce a solution overlap with only two or three layers, which is less than the minimum required for convergence nl < 2n/n. For five qubits, it delivered reduced solution and norm errors compared to other ansätze, independent of the number of layers. In this benchmark, the additional term introduced by the Dirichlet boundary condition does not diminish the superior performance of the full circular ansatz.
2.4 Initialization
An initial quantum state |u(0)〉 can be prepared through classical optimization and accepting converged solutions whose norms fall below a specified threshold (Fontanela et al., 2021; Alghassi et al., 2022) or direct encoding using quantum generative adversarial networks (Zoufal et al., 2019). In most cases, quantum encoding is cost-prohibitive, and sub-exponential encoding can be achieved only under limiting conditions (Nakaji et al., 2022; Mitsuda et al., 2022).
The Dirac delta function is a popular initial probability distribution found in Fokker–Planck equations (Kubo et al., 2021; Alghassi et al., 2022). To encode the state |x〉 in the computational basis
It turns out that for a full circular ansatz [Figure 1(d)], encoding |x〉 does not necessarily require costly optimization. To access a given state |x〉, one can search for a parameterized layer nl – k such that a π bit-flip rotation on an
Figure 3 shows that for a four-qubit full circular ansatz, all 24 – 1 = 15 |x〉 states can be encoded by a single π bit-flip rotation of an Ry gate within four parameterized layers.
2.5 Time complexity
To assess the time complexity of the VQS algorithm, we estimate the number of quantum circuits required per time-step as:
For each circuit, the time complexity scales as (Sato et al., 2021):
To solve an evolution PDE [e.g. equation (1)], a classical algorithm iterates a matrix of size 2n × 2n, compared to a np × np matrix for VQS, suggesting comparable performance at nl ≈ 2n/n.
3. Colloidal transport
With the VQS framework in place, one can explore applications in solving PDEs, such as heat, Black–Scholes and Fokker–Planck equations listed in Alghassi et al. (2022). In this study, we focus on colloidal transport as an application of choice, as the governing Smoluchowski equation involves deep interaction potential energy wells which can be modelled as a component of the Hamiltonian operator [equation (9)], an aspect oft-neglected in quantum simulations.
3.1 Smoluchowski equation
Consider a spherical colloidal particle of radius a near a planar wall (Torres-Díaz et al., 2019). The generalized Smoluchowski equation (Smoluchowski, 1916) describes the probability p(h, t) of locating the particle at h, the distance of the particle centre from the wall at time t, as:
With that, the first and second derivatives of the interaction energy in separation distance are:
Rescaling time τ = tD/a2, we rewrite equation (20) in dimensionless form, which gives the evolution of the probability p(τ) = p(H, τ) as:
Substituting p(τ) = ρ(τ)e–U/2 (Fontanela et al., 2021), we express in Dirac notation:
and the potential term:
3.1.1 Potential-free case (φ = 0).
Consider first the potential-free case where colloid–wall interactions are absent (φ = 0). The probability density state evolves in space H ∈ [0, 1] and time τ ∈ [0, T] as:
To assess the costs of over-parameterization, we calculate the mean trace error [Figure 4(c)] and norm error [Figure 4(d)] depending on the total number of circuits required Nq for the VQS with a run-time of T = 10−1. Figure 4(c) shows that the mean trace error is insensitive to number of ansatz layers up to six and time-steps up to 5 × 10−4; it is reduced only with further increase in the number of ansatz layers nl > 6, leading to optimal scaling of
Using Euler time-stepping, the mean norm error scales as
3.1.2 DLVO potential φ(A, Z, κ).
In the presence of colloid–wall interactions, the DLVO potential term φ depends minimally on three parameters, specifically A, Z and κ [equation (22)]. Following the potential-free case (n = 4, H ∈ [0, 1], Δτ = 10−4), we perform VQS including φ using eight ansatz layers in time τ ∈ [0, T].
In the absence of the electric double layer (Z = 0), the DLVO potential φ(A) depends on only the van der Waal’s interaction energy, assumed here to be attractive. Figure 5(a) shows that the DLVO potential φ(H) profiles scaled by the square of the interval ΔH2 for A ∈ {0.05, 0.1, 0.2, 0.5} is only short-ranged in H, so the quantum solution |ρ〉 is insensitive to A. Recall, however, the earlier substitution p(τ) = ρ(τ)e–U/2, such that the actual solution p depends on the longer-ranged interaction energy U(H) [equation (21)], as shown in Figure 5(a, inset). Indeed, space-time plots show that the colloidal probability density p(H, τ) for A = 0.05 up to T = 0.1 is depleted near wall [Figure 5(c)] compared to the potential-free (φ = 0) case [Figure 5(b)]. Increasing A further increases the depletion range [Figure 5(d)].
Otherwise, the DLVO potential φ(A, K, κ) includes the electric double layer interaction energy, assumed here to be repulsive. For A = 0.5, Figure 6(a) shows that the DLVO potential shows short-ranged dependence on Z and κ. However, p depends on the longer-ranged interaction energy U(H) that can be either attractive or repulsive as shown in Figure 6(a, inset). A space-time plot of the colloidal probability density p(H, τ) for {Z, κ} = {10, 10} shows long-ranged influence of the electric double-layer interaction. Parametric analyses of {Z, κ} holding A = 0.5 show that Z depletes p(H, τ) near wall [Figure 6(c)], and a decrease in κ increases the deposition flux and depletion range [Figure 6(d)].
3.1.3 Trace and norm errors.
Here, we characterize the effect of DLVO potential on the solution fidelity in time using the trace error εtrace(τ) [equation (15)] and the norm error εnorm(τ) [equation (16)]. Figure 7 shows that εtrace(τ) peaks and decreases during the early diffusion phase [Figure 4(a)], then peaks and decreases again as the normalized probability density
Thus concludes our analysis of the potential term in equation (28) in Smoluchowski equation. What usually follows are calculations of survival probability, the probability that the colloidal particle has not reached the wall, the mean first passage time distribution and the mean rate of change of survival probability. Because they do not involve any quantum computation, they are outside the scope of this study. Interested readers are referred to Torres-Díaz et al. (2019).
3.2 Einstein–Smoluchowski equation
The general PDE introduced in equation (1) includes a non-homogeneous source term f, which is not admissible in Smoluchowski’s description of colloidal probability density. To explore the effects of a source term, we switch over to the analogous Einstein–Smoluchowski equation (Cejas et al., 2019; Leong et al., 2023):
3.2.1 Initialization.
We seek a parameterized ansatz that encodes a Heaviside step function centred at |2n–1〉:
For a full circular ansatz [Figure 1(d)], this can be encoded on a minimum of two RY parameterized layers by setting the final layer as
3.2.2 Solutions and errors.
We perform VQS on a 2n = 16 grid using time-step Δτ = 10−4 as before, but on a full circular ansatz with five layers, which is already shown to yield high-fidelity solutions [Figure 2(c) and 2(d)]. Figure 8(a) shows how the normalized concentration
4. Conclusion
Currently, neither VQO nor simulation is capable of realizing an advantage for solving PDEs over classical methods (Anschuetz and Kiani, 2022), but that gap is closing fast (Tosti Balducci et al., 2022). For VQS, a significant progress has been made since the advent of imaginary time evolution (McArdle et al., 2019) notably in the field of quantum finance (Fontanela et al., 2021; Miyamoto and Kubo, 2021; Kubo et al., 2021).
Here, we list a formal approach to solving a 1D evolution PDE [equation (1)]:
{∂tu(t), ∂xxu(t)} terms handled using variational quantum imaginary time evolution.
∂xu(t) term eliminated through substitution methods, such as u(t) = egv(t).
u(t) term included in the Hamiltonian
f(t) term realized by an additional set of complementary circuits, whose complexity depends on
Superior performance of VQS is contingent on two factors: selection of ansatz and initialization of parameters. Comparing real-amplitude ansätze (Section 2.3), we found that the full circular ansatz significantly outperformed not only linear entangled ansätze but also the popular circularly entangled ansatz but with the final parameterized layer unentangled (Kubo et al., 2021; Alghassi et al., 2022). The advantage in solution fidelity persists over multiple parametric layers, which suggests that unentangled parameterized gates reduce overlap with quantum states that are characteristic of PDE solutions. For an initial state resembling a Dirac delta function (Section 2.4), we found that full circular ansatz can be mapped parametrically to a desired state |x〉, thus reducing subsequent impulse encodings to only a trivial lookup.
As a proof-of-concept, we performed VQS to simulate the transport of colloidal particle to an absorbing wall as described by the Smoluchowski equation (Section 3.1) and found high solution fidelity during the initial spreading of the probability distribution. However, to satisfy the asymmetric boundary conditions, additional parameter layers are required, for example, up to —six to eight layers for a four-qubit problem. Higher-order time-stepping such as Runge–Kutta method can reduce norm errors more effectively than over-parameterization for the same time complexity.
With near-wall DLVO potentials, we found that the van der Waal’s interaction impacts VQS mainly through the potential φ(A) of the Hamiltonian, whereas the electric double layer interaction affects the solution mainly through the factor e–U/2 obtained from change of variables. Simulations of colloidal concentration with unit boundary source in the far field (Section 3.2) require additional circuit evaluations equal to approximately half the number of parameters. Interestingly, this cost is offset by the fact that fewer parameters are required, here, for example, five layers for a four-qubit problem.
Overall, we find VQS an efficient tool for applications in colloidal transport because DLVO potentials do not incur additional costs in terms of quantum complexity. Compared to VQE (Leong et al., 2022), VQS enjoys significant advantages in that it does not require repeated encodings and iterative optimization loops. In terms of scalability, we found that the accuracy of quantum simulation not only depends on the number of qubits but also on the imposed boundary and the initial conditions. As with other gradient-based neural networks, VQS potentially suffers from barren plateau problems, which are exemplified by vanishing gradients on flat energy landscapes (McClean et al., 2018) and exacerbated by quantum circuits with high expressivity (Holmes et al., 2022). Mitigation strategies for barren plateaus remain an active area of research (Patti et al., 2021).
Future work can include extension to 2D model for non-spherical colloids (Torres-Díaz et al., 2019), optimal ansatz architecture (Tang et al., 2021) and initial state preparation (Nakaji et al., 2022; Zoufal et al., 2020).
Figures
Figure 2.
Initial step evolves under (a) periodic and (b) Dirichlet boundary condition on a 2n = 16 grid for a real-amplitude ansatz with four layers using time-step Δt = 10−4 plotted in increments of 2 × 10−3; mean (c) trace and (d) norm error plotted on a semi-log scale against the number of ansatz layers for various real-amplitude designs under periodic (open symbols) and Dirichlet (closed symbols) boundary conditions up to T = 10−2 (insets show peak error)
Figure 4.
Normalized colloidal probability density
Figure 5.
(a) DLVO potential φ(H) profiles scaled by ΔH2 using a full circular ansatz with eight layers for Z = 0 varying A. Inset shows interaction energy U(H). (b–d) Space-time plots of colloidal probability density p(H, τ) ∈ [Δp, 0.2] with contour interval Δp = 0.01. Given Z = 0, (b) φ = 0, (c) A = 0.05 and (d) A = 0.5
Figure 6.
(a) DLVO potential φ(H) profiles scaled by ΔH2 using a full circular ansatz with eight layers with A = 0.5 varying Z and κ. Inset shows interaction energy U(H). (b–d) Space-time plots of colloidal probability density p(H, τ) ∈ [Δp, 0.2] with contour interval Δp = 0.01. Given A = 0.5, {Z, κ} are (b) {10, 10}, (c) {20, 10} and (d) {10, 5}
Figure 8.
Normalized colloidal concentration
Figure A1.
Quantum circuits to evaluate (a)
Notes
For an introduction to quantum computation and Dirac notation, we refer the reader to Nielsen and Chuang (2010).
Not to be confused with the cyclic shift operator in equation (12), also denoted by S.
Appendix 1. Quantum circuits to evaluate A and C
The elements of matrix A [equation (6)] and vector C [equation (7)] can be evaluated via sampling the expectation of an observable Z using quantum circuits shown in Figure A1 (Zoufal et al., 2021). The derivative of the trial state
We implemented Hadamard tests [Figure 1(a)] in IBM Qiskit using the aer_simulator backend with sampling count of 212 shots per circuit evaluation and direct measurements using statevector_simulator with respect to observables
Appendix 2. On the encoding of bit strings using full circular ansatz
In this appendix, we elaborate on the initialization procedure described in Section 2.4 that uses the full circular ansatz of Figure 1(d). Firstly, we note that the ansatz is diagonal in the computational basis and therefore preserves computational basis states. Hence, for the rest of this analysis, it suffices to just consider the action of the ansatz on bit strings. As a function Cn: {0, 1}n → {0, 1}n, the ansatz transforms bit strings as follows in little-endian:
For example, when n = 6, one can check that:
Hence, Cn can be represented as the n × n matrix whose (i, j)-th entry is (Cn)ij = 0 if j > i > 0 or i = j =0, and 1 otherwise. For example, when n = 6, this matrix is given by:
Each layer of entangling CNOT gates (Figure 1) corresponds to an application of the Cn matrix to the input bit string, denoted |x0〉. Successive application of Cn generates a sequence of bit strings
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Acknowledgements
This work was supported in part by the Agency for Science, Technology and Research (A*STAR) of Singapore (#21709) under Grant No. C210917001. F.Y.L. acknowledges funding support from the RIE2020 Advanced Manufacturing and Engineering (AME) IAF-PP Capsule Surface Affinities (Complete Life-Cycle) grant (Project Ref.: A20G1a0046). D.E.K. acknowledges funding support from the A*STAR Central Research Fund (CRF) Award for Use-Inspired Basic Research (UIBR), as well as the National Research Foundation, Singapore, and A*STAR under the Quantum Engineering Programme (NRF2021-QEP2-02-P03).
The authors acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team.
Author contributions: F.Y.L. designed study; D.E.K. and J.F.K. advised study; F.Y.L. and W.B.E. wrote software code; F.Y.L. and D.E.K. ran simulations and analysed data. All authors wrote and reviewed the manuscript.
Competing interests: The authors declare no competing interests.
Availability of data and materials: The data sets used and/or analysed during the current study are available from the corresponding author on reasonable request.