Escaping barren plateaus in approximate quantum compiling Niall F. Robertson1 Albert Akhriev1 Jiri Vala23and Sergiy Zhuk1 1IBM Quantum IBM Research Europe - Dublin IBM Technology Campus Dublin 15 Ireland

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Escaping barren plateaus in approximate quantum compiling

Niall F. Robertson1, Albert Akhriev1, Jiri Vala2,3and Sergiy Zhuk1

1IBM Quantum, IBM Research Europe - Dublin, IBM Technology Campus, Dublin 15, Ireland

2Maynooth University, Maynooth, Ireland

3Tyndall National Institute, Cork, Ireland

Abstract

Quantum compilation provides a method to translate quantum algorithms at a high level of

abstraction into their implementations as quantum circuits on real hardware. One approach to

quantum compiling is to design a parameterised circuit and to use techniques from optimisation to ﬁnd

the parameters that minimise the distance between the parameterised circuit and the target circuit of

interest. While promising, such an approach typically runs into the obstacle of barren plateaus - i.e.

large regions of parameter space in which the gradient vanishes. A number of recent works focusing

on so-called quantum assisted quantum compiling have developed new techniques to induce gradients

in some particular cases. Here we develop and implement a set of related techniques such that

they can be applied to classically assisted quantum compiling. We consider both approximate state

preparation and approximate circuit preparation and show that, in both cases, we can signiﬁcantly

improve convergence with the approach developed in this work.

1 Introduction

Quantum compiling is a set of tools to translate a quantum circuit into a set of gates that can be eﬃ-

ciently implemented on quantum hardware. To do so, one must take into account hardware constraints

such as connectivity and maximum circuit depth. Limited hardware connectivity between qubits requires

additional operations such as SW AP gates to be included into the compilation process [1]. Similarly,

the physical implementation of entangling two-qubit operations such as a controlled-NOT (CN OT ) op-

eration, illustrated in Fig. 1, may require that only one of the two qubits can play the role of the control

qubit. Despite recent progress on error mitigation techniques [2–4], noise on present day devices signif-

icantly limits the depth of the circuit that can be implemented - the beneﬁt of optimally decomposing

a quantum circuit into the hardware’s native gate alphabet and connectivity constraints is hence self-

evident. We can classify quantum compiling techniques into two broad categories: classically assisted and

quantum assisted. In the former case, the idea is to use classical methods to compile sub-circuits that

form part of a larger quantum algorithm. The main advantage of such an approach is that one can avoid

the issues arising from noise on a quantum device, the disadvantage being that, due to the exponential

scaling of the Hilbert space, one is limited to sub circuits that act on a ﬁxed number of qubits. The

quantum assisted approach however is scalable and can hence be applied to the entire quantum circuit of

interest. However, one encounters quantum noise which, even in cases where the compilation scheme is

somewhat noise resilient [5], can cause signiﬁcant convergence issues [6]. Furthermore, the circuits that

have been proposed to implement a quantum assisted quantum compilation procedure require long range

qubit connectivity on the quantum device of interest [7].

Here we focus on the classically assisted approach to quantum compilation. Classically assisted vari-

ational quantum algorithms have been recently studied in [8,9]. In these works, Tensor Networks are

used to generate a shallow circuit with weak entanglement. The circuit is subsequently deepened on

a quantum device, and the entanglement is thus increased beyond what classical techniques can store.

These recent promising results provide signiﬁcant motivation to improve the state of the art of classical

optimisation techniques applied to quantum circuits which are currently limited by convergence issues

such as barren plateaus, as will be discussed below.

arXiv:2210.09191v1 [quant-ph] 17 Oct 2022

We consider the problem of approximate quantum compiling as studied in [10]. This approach involves

the design of a parametric quantum circuit with ﬁxed depth - the parameters are then adjusted to bring it

as “close” as possible to the target, where “close” is deﬁned via some carefully chosen metric (see section

2.2). In [10], it was shown that such an approach could be used to derive a 14-CNOT decomposition of a

4-qubit Toﬀoli gate from scratch. It was furthermore shown that the Quantum Shannon Decomposition,

introduced by Shende, Bullock and Markov [11], can be compressed by a factor of two without practical

loss of ﬁdelity. In this work, our main contribution is a new classical algorithm for quantum compilation

which signiﬁcantly improves convergence - see below and main text for details.

In addition to the sub-circuit problem discussed above, we consider state preparation as an example

where classical techniques can play a signiﬁcant role. For a large class of quantum algorithms, the en-

tanglement of the wavefunction is weakly entangled at the beginning of the circuit and grows with the

depth. There exist widely studied classical techniques (i.e. Tensor Networks) to store the wavefunction

of a large number of qubits when the system is only weakly entangled. Thus, this opens the possibility

of using classical Tensor Network techniques for state preparation to optimally compile the early parts

of a quantum circuit, in a similar way to [8,9]. One of the biggest obstacles however to any quantum

compilation technique, whether it is quantum assisted or not, is the existence of barren plateaus in the

parameter landscape, i.e. large regions where the gradient of the cost function vanishes exponentially in

the number of qubits, thus rendering it extremely diﬃcult to train. A proposed solution for the quantum

assisted quantum compiling approach is to use shallow circuit Ansatz’ and so-caled “local cost functions”.

In short, a cost function calculated via a quantum circuit is local if it involves the measurement of only a

small number of qubits - a global cost function involves the measurement of all qubits. Here we design a

new classical algorithm inspired by the distinction between local and global cost functions made in [7,12]

for quantum compilation. In particular, we derive an expression inspired by the quantum approach that

can be easily implemented on a classical computer and that allows us to escape the barren plateaus. This

new cost function introduces a number of coeﬃcients that we must update throughout the optimsation

procedure. Our algorithm implements a scheme to do so eﬀectively, such that short depth parametric

quantum circuits can be found that closely approximate the target circuit of interest.

Our main contributions in this work are as follows: we derive an explicit expression for the local cost

functions considered in [7,12] in equation (14), such that they obtain a clear and intuitive operational

meaning. We use this to develop a new cost function - see equation (21) - whose gradient can be eﬃciently

calculated classically. We consider a particular case where our cost function can be studied analytically

and we show how the variance of its gradient behaves - see equation (29) - which allows us to understand

how and why our classical compilation scheme allows for the escape from barren plateaus. We develop

an algorithm to update the coeﬃcients appearing in our new cost function - see section 4, appendix A

and appendix B. Finally, we demonstrate numerically in Figures 6and 8that the combination of our cost

function and our algorithm to update the coeﬃcients signiﬁcantly improves the convergence of certain

problems in quantum compilation.

The outline of this article is as follows: in section 2, we review approximate quantum compiling and

we show how it can be formulated as an optimisation problem. In section 3, we discuss the convergence

issues that are commonly encountered in quantum compiling problems - we discuss in particular the

appearance of barren plateaus in the parameter landscape. We introduce our classical version of the

local cost function and we show how, for a particular example where the cost function can be studied

analytically, the variance of the gradient of our cost function behaves as a function of the number of

qubits. In section 4, we describe our scheme to update the coeﬃcients that appear in our classical local

cost function over the course of the optimisation procedure. In section 5, we present some numerical

results on simulations for up to 24 qubits showing that our classical local cost function signiﬁcantly

improves convergence as compared to the global one. We conclude in section 6.

2 Setup

Quantum compilation is an active research area with a wide spectrum of research directions. One such

direction is to ﬁnd the the best representation - in terms of some distance or structural properties - of a

generic unitary in a certain canonical basis (e.g. [13–17]). One can also consider an approach based on

exact qubit routing, i.e. mapping logical qubits to physical qubits so that the resulting unitary coincides

with the original one (up to a permutation) while allowing for compatibility with the hardware topology

(e.g. [18–31]). As discussed in the introduction, the approach to quantum compilation that we focus on

here is to design a parametric quantum circuit with ﬁxed depth and to adjust the parameters of this

circuit to bring it as “close” as possible to the target, where “close” is deﬁned via some carefully chosen

metric. While one can in principle use universal parametric circuits which provide a theoretical guarantee

that there exist parameters such that any target circuit can be exactly compiled, such circuit templates

result in impractically deep circuits for present day quantum devices. Approximate quantum compiling

can be seen as a compilation approach that trades oﬀ the circuit accuracy for its depth. It was shown

in [32] that generic unitary operations over nqubits are computationally hard even to approximate. More

recently it has been shown that quantum circuit compilation is an NP-complete problem [33]. However,

the fact that there is a subset of unitary operations that can be realized in quantum circuits of lower

complexity, such as those in the Shor algorithm [34] for example, suggests that quantum compilation

may oﬀer both an interesting perspective on complexity of quantum circuits and perhaps a path to new

quantum algorithms that would provide quantum advantage.

Figure 1: Controlled-NOT (CN OT ) operations with the role of a control qubit played by the ﬁrst and

second qubit respectively. CNOT transforms each standard computational basis element in a quantum

superposition by adding a value of the control qubit to that of the target qubit modulo 2: |aic⊗ |bit→

|aic⊗ |b⊕amod 2itwhere a, b ∈ {0,1}.

2.1 Parametric quantum circuits and ansatz

As discussed in [10], a natural circuit Ansatz is one based on so-called CNOT blocks. A CN OT block is

aCNOT gate followed by single qubit rotations (see Figure 2). Single-qubit rotation gates are deﬁned

as complex exponential functions of the Pauli matrices:

Rx(θ) = exp(−iσxθ/2) = cos(θ/2) −isin(θ/2)

−isin(θ/2) cos(θ/2) ,

Ry(θ) = exp(−iσyθ/2) = cos(θ/2) −sin(θ/2)

sin(θ/2) cos(θ/2) ,

Rz(θ) = exp(−iσzθ/2) = exp(−iθ/2) 0

0 exp(iθ/2) .

(1)

A block with a CNOT gate acting on a “control” qubit jand “target” qubit kis written as CUjk(θ1, θ2, θ3, θ4).

Ry(θ1)Rz(θ2)

Ry(θ3)Rx(θ4)

Figure 2: CNOT block forms the basic building block of our circuit ansatz.

For a given hardware connectivity, one can then write down a full parameterised circuit as:

Vct(θ) =CUct(L)(θ3n+4L−3, . . . , θ3n+4L)· · · CUct(1) (θ3n+1, . . . , θ3n+4)

[Rz(θ1)Ry(θ2)Rz(θ3)] ⊗ · · · ⊗ [Rz(θ3n−2)Ry(θ3n−1)Rz(θ3n)] (2)

The position of the CNOT blocks in the parameterised circuit can be customised to suit the particular

target circuit that one is interested in. We will use a particular structure of the ansatz which is illustrated

in Figure 3. In this circuit Ansatz, we have 3 rotation angles for each qubit at the beginning, we then

have llayers in which we repeat each CN OT block btimes. In Figure 3, we have n= 4 qubits, l= 2 and

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EscapingbarrenplateausinapproximatequantumcompilingNiallF.Robertson1,AlbertAkhriev1,JiriVala2;3andSergiyZhuk11IBMQuantum,IBMResearchEurope-Dublin,IBMTechnologyCampus,Dublin15,Ireland2MaynoothUniversity,Maynooth,Ireland3TyndallNationalInstitute,Cork,IrelandAbstractQuantumcompilationprovidesamethodtot...

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Escaping barren plateaus in approximate quantum compiling Niall F. Robertson1 Albert Akhriev1 Jiri Vala23and Sergiy Zhuk1 1IBM Quantum IBM Research Europe - Dublin IBM Technology Campus Dublin 15 Ireland

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