Algorithms for perturbative analysis and simulation of quantum dynamics Daniel Puzzuoli1 Sophia Fuhui Lin2 Moein Malekakhlagh3 Emily Pritchett3

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Algorithms for perturbative analysis and simulation of

quantum dynamics

Daniel Puzzuoli1,*, Sophia Fuhui Lin2, Moein Malekakhlagh3, Emily Pritchett3,

Benjamin Rosand3, and Christopher J. Wood3

1IBM Quantum, IBM Canada, Markham, ON, L3R 9Z7, Canada

2Department of Computer Science, University of Chicago, Chicago, IL, 60615, USA

3IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA

*daniel.puzzuoli1@ibm.com

June 7, 2023

Abstract

We develop general purpose algorithms for computing and utilizing both the

Dyson series and Magnus expansion, with the goal of facilitating numerical pertur-

bative studies of quantum dynamics. To enable broad applications to models with

multiple parameters, we phrase our algorithms in terms of multivariable sensitivity

analysis, for either the solution or the time-averaged generator of the evolution over

a ﬁxed time-interval. These tools simultaneously compute a collection of terms up

to arbitrary order, and are general in the sense that the model can depend on the

parameters in an arbitrary time-dependent way. We implement the algorithms in

the open source software package Qiskit Dynamics, utilizing the JAX array library

to enable just-in-time compilation, automatic diﬀerentiation, and GPU execution of

all computations. Using a model of a single transmon, we demonstrate how to use

these tools to approximate ﬁdelity in a region of model parameter space, as well as

construct perturbative robust control objectives.

We also derive and implement Dyson and Magnus-based variations of the re-

cently introduced Dysolve algorithm [Shillito et al., Physical Review Research,

3(3):033266] for simulating linear matrix diﬀerential equations. We show how the

pre-computation step can be phrased as a multivariable expansion computation

problem with fewer terms than in the original method. When simulating a two-

transmon entangling gate on a GPU, we ﬁnd the Dyson and Magnus-based solvers

provide a speedup over traditional ODE solvers, ranging from roughly 2×to 4×for

a solution and 10×to 60×for a gradient, depending on solution accuracy.

1 Introduction

Accurate and high-performance simulation of the physics of quantum systems is a key

component in device and control engineering workﬂows in quantum computation. Due

arXiv:2210.11595v2 [quant-ph] 6 Jun 2023

to the complexity of the time-dependent diﬀerential equations involved, perturbative

techniques are commonly employed to simplify models of these systems. The Dyson

series [1] and Magnus expansion [2, 3] are widely used time-dependent perturbation

theory tools for studying quantum system dynamics. As perturbative expansions, they

enable sensitivity analysis: the quantiﬁcation of how the dynamics change with local

changes to system parameter values. They are utilized heavily in theoretical studies of

quantum systems, as they enable perturbative extension of analytically tractable prob-

lems through the construction of “eﬀective Hamiltonians” [4, 5, 6, 7]. For example, in

superconducting quantum computing, eﬀective Hamiltonian models have been derived

for microwave-activated single-qubit [8] and two-qubit gates, such as the cross-resonance

gate [9, 10]. In the ﬁeld of quantum control, starting with Average Hamiltonian Theory

[11], these expansions have also been used to design open-loop robust control sequences

that suppress the eﬀect of perturbations [12, 13, 14, 15, 16, 17, 18], sequences for sensing

by enhancing perturbations [19], dynamical decoupling [20, 21, 22], dynamically cor-

rected gates [23, 24], the ﬁlter function formalism [25, 26], and Magnus-based methods

for suppressing non-adiabatic errors [27], among others.

More recent work has focused on the numerical computation of these expansions,

often with the goal of generalizing the previously mentioned applications to contexts

which are not analytically tractable. For robust control, numerical methods have been

developed to compute expansion terms in the interaction frame [28, 11, 15] of a control

sequence, e.g. the Dyson-like terms of [29, 30] and ﬁlter function formalism terms [31,

32, 33], the latter of which have been implemented in software packages [34, 35]. In the

context of simulation of quantum systems, Ref. [36] gives an algorithm for computing

Dyson-series-derived expressions as part of the pre-computation step of the Dysolve

algorithm, with similar methods appearing in [37].

In this paper we develop software tools for numerically computing and utilizing

the Dyson series and Magnus expansion. The general goal is to deﬁne and implement

algorithms for broad versions of these computational problems, so that they may be

used as primitives in many numerical research applications, including robust control

and classical simulation of quantum systems. By making the construction of these terms

easily accessible, their usefulness in new and existing methods can be explored to higher

orders and applied to models with more parameters.

Towards this end, this paper is organized as follows:

•Section 2 deﬁnes the multivariable Dyson series and Magnus expansion, and gen-

erally introduces the power series and interaction frame notation used throughout

the paper.

•Section 3 gives algorithms for computing both multivariable Dyson series and Mag-

nus expansion terms, analyzes their scaling, and describes the implementation in

the Qiskit Dynamics package.

•Section 4 demonstrates the usage of the software tools in an example of a robust

control problem. In particular, it is shown how the tools can be used to approx-

imate gate ﬁdelity in a region of model parameter space, as well as to construct

robustness objectives incorporating higher order expansion terms.

•Section 5 derives Dyson and Magnus-based variants of the Dysolve algorithm [36]

for solving linear matrix diﬀerential equations, and describes an implementation of

these algorithms in Qiskit Dynamics, building on the implementation described in

Section 3. We benchmark these solvers against the standard ODE solvers available

in Qiskit Dynamics for the problem of simulating a two transmon CR gate. We

ﬁnd a speed up over traditional solvers in Qiskit Dynamics of roughly 2×to 4×

for a solution and 10×to 60×for a gradient on GPU, depending on the accuracy

of the solutions.

The paper ends with a discussion in Section 6.

2 Multivariable Dyson series and Magnus expansion

We begin by informally introducing the computational problems we consider in this

paper. Consider a linear matrix diﬀerential equation (LMDE):

U(t, c0, . . . , cr−1) = G(t, c0, . . . , cr−1)U(t, c0, . . . , cr−1),(1)

where tis time, and c0, . . . , cr−1are some parameters of the generator Gand solution U,

which are assumed to be perturbative. Note that the Schrodinger equation is an LMDE

under the association G=−iH, for Hthe Hamiltonian, along with other common

master equations, such as the Lindblad and Bloch-Redﬁeld equations.1

Given an integration interval [t0, tf] and a power-series decomposition of Gin the

parameters c0, . . . , cr−1(centred at 0):

G(t, c0, . . . , cr−1) = G∅(t) +

∞

k=1 X

0≤i1≤···≤ik≤r−1

ci1. . . cikG(i1,...,ik)(t),(2)

with the functions G∅and G(i1,...,ik)being completely arbitrary user-deﬁned functions,

the problem is to compute corresponding terms in the truncated power series for the

solution Uitself:

U(t0, tf) = U∅+

∞

k=1 X

0≤i1≤···≤ik≤r−1

ci1. . . cikU(i1,...,ik),(3)

or for the time-averaged generator Ω:

Ω=Ω∅+

∞

k=1 X

0≤i1≤···≤ik≤r−1

ci1. . . cikΩ(i1,...,ik),(4)

1The Lindblad and Bloch-Redﬁeld equations, which are diﬀerential equations for density matrix

evolution, are not typically presented in the form of Equation (1). However, their right-hand sides are

linear functions of the density matrix, and therefore they can be rewritten in the form of Equation (1)

using a vectorization convention.

implicitly deﬁned to satisfy U(t0, tf) = exp(Ω).2

While we phrase the time-dependent operators G∅(t) and G(i1,...,ik)(t) in Equation

(2) as being arbitrary from a computational generality perspective, they are ﬁxed by

the parameterization of the generator G(t, c0, . . . , cr−1). That is, it holds that G∅(t) =

G(t, 0,...,0) (i.e. G∅(t) is the unperturbed generator), and each G(i1,...,ik)(t) is the par-

tial derivative of G(t, c0, . . . , cr−1) with respect to the variables ci1, . . . , cik(up to the re-

quired combinatorial pre-factor for multivariable Taylor series). Hence, they are entirely

determined by how Gis written in terms of the perturbative parameters c0, . . . , cr−1.

Lastly, we note that the formal algorithms in Section 3 solve the above problem in

the interaction frame of G∅(t)[28, 11, 15], which is reviewed in Section 2.2.

2.1 Power series notation

Using Equation (2) as a model, we introduce notation to simplify working with mul-

tivariable power series. Each term in Equation (2) is indexed by a list of indices

0≤i1≤ ··· ≤ ik≤r−1. What uniquely identiﬁes the above term is the number

of times each index in {0, . . . , r −1}appears, and hence a multi-index notation [38] is

often used for multivariable power series. We use a slightly diﬀerent, though function-

ality equivalent, notation in terms of multisets. A multiset is like a set, but in which

repeated elements may appear.

Denoting multisets with round brackets, the multiset associated with the above power

series term is given as: I= (i1, . . . , ik). Given a list of variables c0, . . . , cr−1, we denote

cI=ci1× ··· × cik.(5)

For example, c(0,0,1) =c2

0c1, and c(0,1,1,2) =c0c2

1c2. This notation enables a simple

correspondence between algebraic operations and multiset operations. E.g. for two

multisets I, J, we have that:

cI+J=cI×cJ,(6)

where I+Jdenotes the multiset summation.

With this, for I= (i1, . . . , ik), the summation term ci1. . . cikG(i1,...,ik)(t) in Equation

(2) is rewritten as

cIGI(t).(7)

Letting Ik(r) denote the set of multisets of size kwith elements in {0, . . . , r −1}, and

letting cgenerally denote a list of variables (c0, . . . , cr−1), we may rewrite Equation (2)

G(t, c)≡G(t, c0, . . . , cr−1) = G∅(t) +

∞

k=1 X

I∈Ik(r)

cIGI(t).(8)

Lastly, we use |I|to denote the number of elements in the multiset, including repeats.

For example, |(0,0,1)|= 3. This notation is used throughout the paper to represent

power series.

2These tasks can equivalently be phrased as performing sensitivity analysis, or numerically ﬁnding a

series solution, albeit specialized to the case of a linear matrix diﬀerential equation.

2.2 Interaction frame

In many quantum control applications, computations are performed in the interaction

frame of the unperturbed generator [28, 11, 15]. The beneﬁt of the interaction frame is

that it factorizes the evolution generated by G(t, c) into two pieces: one given purely by

G∅(t), and the other being trivial if c= 0. Denoting

V(t) = Texp Zt

dsG∅(s),(9)

with Tbeing the time-ordering operator [39], the generator Gin Equation (8), trans-

formed into the interaction frame of G∅, is given by:

G(t, c) =

∞

k=1 X

I∈Ik(r)

cI˜

GI(t),(10)

where ˜

GI(t) = V−1(t)GI(t)V(t). With this, it holds that:

Texp Zt

dsG(s, c)=V(t)Texp Zt

ds ˜

G(s, c).(11)

2.3 Multivariable Dyson series

For a generator G(t), the Dyson series [1] expands

Texp Zt

dsG(s)=1+

∞

k=1

Dk(t),(12)

where 1is the identity operator, and for the explicit formula

Dk(t) = Zt

dt1···Ztk−1

dtkG(t1). . . G(tk).(13)

We may view this as a power series in a single variable cby using this formula for the

generator cG(t):

Texp cZt

dsG(s)=1+

∞

k=1

ckDk(t),(14)

where the Dk(t) are still as in Equation (13).

To deﬁne the multivariable Dyson series, the goal is to generalize the above equation

and explicit expression in Equation (13) to an arbitrary number of variables, with the

original generator given as a power series.

Deﬁnition 1. Let ˜

G(t, c) be as in Equation (10), with crepresenting a list of variables.

The multivariable Dyson series is the power series of the solution in c:

Texp Zt

ds ˜

G(s, c)=1+

∞

k=1 X

I∈Ik(r)

cIDI(t),(15)

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AlgorithmsforperturbativeanalysisandsimulationofquantumdynamicsDanielPuzzuoli1,*,SophiaFuhuiLin2,MoeinMalekakhlagh3,EmilyPritchett3,BenjaminRosand3,andChristopherJ.Wood31IBMQuantum,IBMCanada,Markham,ON,L3R9Z7,Canada2DepartmentofComputerScience,UniversityofChicago,Chicago,IL,60615,USA3IBMQuantum,IBMT...

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Algorithms for perturbative analysis and simulation of quantum dynamics Daniel Puzzuoli1 Sophia Fuhui Lin2 Moein Malekakhlagh3 Emily Pritchett3

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