Hybrid Quantum Classical Simulations Dennis Willsch1 Manpreet Jattana12 Madita Willsch13 Sebastian Schulz12 Fengping Jin1 Hans De Raedt14 and Kristel Michielsen123

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Hybrid Quantum Classical Simulations

Dennis Willsch1, Manpreet Jattana1,2, Madita Willsch1,3, Sebastian Schulz1,2,

Fengping Jin1, Hans De Raedt1,4, and Kristel Michielsen1,2,3

1Institute for Advanced Simulation, J¨

ulich Supercomputing Centre, Forschungszentrum J¨

ulich,

52425 J¨

ulich, Germany

E-mail: {d.willsch, m.jattana, m.willsch, se.schulz, f.jin, k.michielsen}@fz-juelich.de

2RWTH Aachen University, 52056 Aachen, Germany

3AIDAS, 52425 J¨

ulich, Germany

4Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747 AG

Groningen, The Netherlands

E-mail: deraedthans@gmail.com

We report on two major hybrid applications of quantum computing, namely, the quantum ap-

proximate optimisation algorithm (QAOA) and the variational quantum eigensolver (VQE).

Both are hybrid quantum classical algorithms as they require incremental communication be-

tween a classical central processing unit and a quantum processing unit to solve a problem. We

ﬁnd that the QAOA scales much better to larger problems than random guessing, but requires

signiﬁcant computational resources. In contrast, a coarsely discretised version of quantum an-

nealing called approximate quantum annealing (AQA) can reach the same promising scaling

behaviour using much less computational resources. For the VQE, we ﬁnd reasonable results in

approximating the ground state energy of the Heisenberg model when suitable choices of initial

states and parameters are used. Our design and implementation of a general quasi-dynamical

evolution further improves these results.

1 Introduction

Quantum computing1is an emerging computer technology that uses quantum effects in the

design of its computational model. There are two major paradigms in quantum computing,

namely the gate-based quantum computer and the quantum annealer.

A gate-based quantum computer is inspired by the circuit model for classical, digital

computers. This means that every program is deﬁned in terms of a sequence of fundamen-

tal operations, the so-called quantum gates. Each quantum gate operates on the fundamen-

tal units of computation, the so-called quantum bits or qubits. The gate-based quantum

computer executes these quantum gates step-by-step and thus updates its internal quantum

state. At the end of the quantum gate sequence, the quantum state is “measured”, which

means that the quantum computer outputs, with a certain probability, one out of several

classical bitstrings. The maximum number of bits in this bitstring is given by the number

of qubits in the quantum computer. At the time of writing, gate-based quantum computers

with approximately one hundred qubits have been manufactured2.

A quantum annealer, on the other hand, tries to harness the natural evolution of a quan-

tum system, steered by external magnetic ﬁelds, to solve an optimisation problem. In-

ternally, it also operates on a set of qubits, which are measured at the end of a quantum

annealing run. It thus also produces a bitstring as output. At the time of writing, quantum

annealers with more than 5000 qubits have been manufactured3.

arXiv:2210.02811v2 [quant-ph] 7 Oct 2022

Bitstrings

Variational

parameters

OUTPUT

Execution Measurement

QUANTUM PROCESSING UNIT

Energy

calculator

Optimiser

CLASSICAL PROCESSING UNIT

|0

Preparation

U1(θ1)

Ut(θt)

...

INPUT

Instructions

Figure 1. Schematic diagram of hybrid quantum classical variational algorithms. The tasks are divided

between a QPU and a CPU. The CPU suggests certain parameters of a parametrised quantum circuit. This

quantum circuit, along with other instructions, is then sent to and executed by the QPU. Following this, the

measurement results from the QPU are sent back to the CPU, which in turn evaluates the energy to be optimised.

From this result, the CPU obtains new parameters for the quantum circuit. This process continues until some

convergence criterion is met.

Both types of quantum computers share the property that the same program can pro-

duce different bitstrings obtained after a run. The computational models are thus inherently

probabilistic. This means that quantum computers can also be used as samplers. A pro-

gram is often executed multiple times to obtain a representative distribution of bitstrings.

In this article, we consider simulations of hybrid quantum classical variational algo-

rithms. Such algorithms obtain their name from the combined usage of gate-based quan-

tum and classical computers in a single application. The working of such algorithms is

visualised in Fig. 1. The optimiser in the classical central processing unit (CPU) sends the

parameters placed in a circuit to be executed by the quantum processing unit (QPU). The

QPU prepares a problem speciﬁc initial state, executes the circuit, and sends the results

to the CPU after measurement. The outcomes of the measurements, called bitstrings, are

used to calculate the energy which is given to the optimiser. The optimiser then decides

what new parameters will lower the energy, and the cycle continues until convergence. The

CPU also controls the QPU through other instructions, e.g. the number of measurements,

microwave pulses, or the time between measurements.

This article is structured as follows. Section 2 contains an overview of the J¨

ulich Uni-

versal Quantum Computer Simulator, which is used for the simulation of the following

two hybrid quantum classical applications. In Sec. 3, we introduce the QAOA, discuss its

relation to AQA, and compare both algorithms when applied to the tail assignment prob-

lem. In Sec. 4, we discuss the VQE and its application to the Heisenberg model. Section 5

contains our conclusions.

Figure 2. Distribution of the coefﬁcients of the state vector |ψiacross the GPUs and compute nodes of

JUWELS Booster7.Each GPU is handled by a single MPI process. For each GPU, the global qubit indices

of the coefﬁcients represent its MPI rank. For the GPUs belonging to MPI rank 0 and 3 this is indicated on the

right (the 10 leftmost indices are the global qubit indices). The complex coefﬁcients of the local qubits are stored

locally on the GPUs.

2 J¨

ulich Universal Quantum Computer Simulator

In this section, we brieﬂy describe the J¨

ulich Universal Quantum Computer Simulator

(JUQCS), and outline how a program for a gate-based quantum computer can be simulated

using supercomputers. More details and in-depth descriptions of the implementation are

given in Refs. 4–6.

The basic unit of computation for quantum computers is the qubit. Mathematically,

a qubit is described by a unit vector of two complex numbers |ψi= (ψ0, ψ1)fulﬁlling

hψ|ψi=|ψ0|2+|ψ1|2= 1. Usually, the basis states (i.e., orthonormal basis vectors) are

denoted by |0iand |1i. An N-qubit system is deﬁned by 2Ncomplex numbers

|ψi=ψ0...00 |0. . . 00i+ψ0...01 |0. . . 01i+. . . +ψ1...11 |1. . . 11i,(1)

where |0. . . 00i,...,|1. . . 11idenote the computational basis states1and the complex

coefﬁcients ψ0...00, . . . , ψ1...11 again fulﬁl hψ|ψi= 1. The 2Ncomplex coefﬁcients in

the state |ψican be written as a rank-Ntensor ψqN−1···q1q0where qj∈ {0,1}denote the

indices.

Since the number of coefﬁcients grows exponentially in the number of qubits N, the

memory requirement grows exponentially too, which makes large-scale simulations of

universal quantum computers only possible on supercomputers with enough (distributed)

random access memory. To simulate for instance a quantum computer with N= 42

qubits, the tensor ψqN−1···q1q0requires 16 ×242 B = 64 TiB of memory (using double

precision ﬂoating-point numbers) while the simulation of N= 21 qubits only requires

16 ×221 B = 32 MiB of memory.

The GPU version of JUQCS (JUQCS–G) distributes the complex coefﬁcients over the

memory of the participating GPUs. A sketch is shown in Fig. 2. Each GPU stores a power

of two coefﬁcients, say 2Mcoefﬁcients, of the state vector |ψiin its local memory. Qubits

jwhose amplitudes ψqN−1...qj+10qj−1...q0and ψqN−1...qj+11qj−1...q0are stored on the same

local memory are thus called local qubits. Qubits whose amplitudes are distributed over

different GPUs are called global qubits. The total number of GPUs needed is NGPU =

2N−M.

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HybridQuantumClassicalSimulationsDennisWillsch1,ManpreetJattana1;2,MaditaWillsch1;3,SebastianSchulz1;2,FengpingJin1,HansDeRaedt1;4,andKristelMichielsen1;2;31InstituteforAdvancedSimulation,J¨ulichSupercomputingCentre,ForschungszentrumJ¨ulich,52425J¨ulich,GermanyE-mail:fd.willsch,m.jattana,m.willsch,s...

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Hybrid Quantum Classical Simulations Dennis Willsch1 Manpreet Jattana12 Madita Willsch13 Sebastian Schulz12 Fengping Jin1 Hans De Raedt14 and Kristel Michielsen123

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