Restoring broken symmetries using oracles Edgar Andres Ruiz Guzmanand Denis Lacroixy Universit e Paris-Saclay CNRSIN2P3 IJCLab 91405 Orsay France

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Restoring broken symmetries using oracles

Edgar Andres Ruiz Guzman∗and Denis Lacroix†

Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

(Dated: October 21, 2022)

We present a new method to perform variation after projection in many-body systems on quantum

computers that does not require performing explicit projection. The technique employs the notion

of “oracle”, generally used in quantum search algorithms. We show how to construct the oracle

and the projector associated with a symmetry operator. The procedure is illustrated for the parity,

particle number, and total spin symmetries. The oracle is used to restore symmetry by indirect

measurements using a single ancillary qubit. An Illustration of the technique is made to obtain the

approximate ground state energy for the pairing model Hamiltonian.

I. INTRODUCTION

Quantum computers promise to speed up the compu-

tation of some selected problems that are hard to solve

on classical computers [1, 2]. One of the opportunities of-

fered by quantum technologies is the simulation of many-

body quantum systems with a large number of particles,

where the exponential scaling of their Hilbert space pre-

vents their ab-initio description in classical computers as

the number of degrees of freedom increases. Assuming we

can describe the physical system accurately in the quan-

tum computer, multiple methods have been designed to

estimate the ground state energy of a Hamiltonian, e.g.,

[3–8]. We are now in the Noisy intermediate-scale quan-

tum (NISQ) computers era [9, 10]; in this period, the al-

gorithms should be tailor-made to handle a limited num-

ber of gates and qubits and the presence of noise. The

variational quantum eigensolver (VQE) [3, 4] is one of the

currently used best candidates to ﬁll the requirements

listed above due (i) to its comparatively short coherence

time and (ii) to the possibility of customizing the ans¨atz

to the physical problem at hand.

Because of the inherent noise of current quantum pro-

cessors, the symmetry that a wave function should re-

spect when solving a physical problem will most likely be

broken accidentally. A possible way to control the errors

is eventually to enforce the symmetry with speciﬁc algo-

rithms [12–14, 16–19]. In some cases, like when a system

encounters a spontaneous symmetry breaking, it can also

be helpful to break some symmetries on purpose [20–24].

In both cases, wanted or unwanted symmetry-breaking

(SB), speciﬁc methods should be designed to restore the

symmetry (SR) in quantum computations. In the many-

body context, this avenue has been recently explored,

requiring [25] or not [26] the explicit construction of the

symmetry projected wave function. The method pro-

posed in [25] applies to any symmetries, including spin

projection problems [27]. The symmetry projection was

used in addition to classical optimization post-processing

calculations in Refs. [28, 29] to obtain ground state

∗Electronic address: ruiz-guzman@ijclab.in2p3.fr

†Electronic address: denis.lacroix@ijclab.in2p3.fr

and excited states in many-body systems. In particular,

in [28], the equivalent to the Variation-After-Projection,

called Q-VAP, has been applied to superﬂuid systems.

This method is based on the Quantum-Phase-Estimation

(QPE) algorithm, using the indirect measurements of a

set of ancillary qubits with a large set of quantum op-

erations to perform in the circuit. This resource de-

mand limited us to only testing the method on quan-

tum emulators, which will probably be usable on quan-

tum platforms after the NISQ period. Alternative meth-

ods to restore symmetries, eventually with lower circuit

depths and lengths, have been discussed recently in Refs.

[30, 31].

One of the promising methods evoked in Ref. [31] is

those based on oracles. Oracles are speciﬁc operators

that have been introduced in quantum search algorithms.

Among these algorithms, one can mention the Grover

method [32–35] that has been recognized as optimal for

speciﬁc query problems [36, 37]. The practical use of or-

acles depends strongly on the diﬃculty of constructing

them. We analyze how oracles can be implemented to

restore symmetries in the Quantum-Variation After Pro-

jection (Q-VAP) method of Ref. [29].

The procedure reduces the cost of the indirect measure-

ments compared to the QPE to that of a single qubit. It

can also continuously monitor symmetry restoration dur-

ing the variational optimization process. In particular, it

can avoid explicitly projecting the variational state each

time the energy is estimated. Illustrations of the oracle

construction are given for the parity, particle number,

and total spin symmetries. Applications are performed

on the pairing model.

II. QUANTUM

VARIATION-AFTER-PROJECTION WITH

ORACLE

Similarly to the Variation-After-Projection (VAP) per-

formed on a classical computer, in the Q-VAP approach,

a symmetry-breaking state |Ψ ({θi})iis considered where

{θi}are a set of parameters that are varied to minimize

arXiv:2210.11181v1 [quant-ph] 20 Oct 2022

the energy:

E({θi}) = hΨ ({θi})|ˆ

Hˆ

PS|Ψ ({θi})i

hΨ ({θi})|ˆ

PS|Ψ ({θi})i.(1)

Here, ˆ

Hdenotes the Hamiltonian, and ˆ

PSis a projector

onto the subspace ˆ

HS, which respects a given set of sym-

metries of the Hamiltonian, denoted generically by S. In

Eq (1), we used the fact that ˆ

S=ˆ

PSand hˆ

H, ˆ

PSi= 0.

The method proposed in Ref [25] to perform symme-

try restoration combines the QPE approach and takes

advantage of the fact that the eigenvalues of symmetry

operators are known. Although this method applies to

any symmetry problem, one of its inconveniences is that

it relies on two steps to estimate the energy in Eq. (1). In

the ﬁrst step, the SR state is obtained by projection, and

in the second step, this projected state is used to compute

the expectation value of ˆ

H. Possible ways to reduce the

circuit to perform the projection have been discussed in

Ref. [31]. Depending on the projection method, this can

lead to quite signiﬁcant coherence time requirements. In

most of the techniques discussed in [31], the projection

is achieved by indirect measurements of a set of ancillary

qubits. Because of this, and despite possible reductions

in the number of operations to perform the projection,

the ﬁrst step often remains probabilistic. This implies

that only part of the events and, depending on the un-

projected state properties, a possibly signiﬁcant fraction

of the runs could be thrown away, leading to waste in

the use of quantum platforms. An extreme situation

would be where the SB state has a very small or zero

fraction of states belonging to HS. Thus, the probabil-

ity of the states with the correct symmetry is too low

or zero in the parametric symmetry-breaking wave func-

tion. In that case, we could ﬁnd ourselves in a situation

where the projector rarely or never projects in the correct

subspace. To address this issue, we explore below an al-

ternative method where we can directly access the energy

in Eq (1) without performing the explicit projection.

A. Using oracle for symmetry restoration

An oracle [31–33], denoted hereafter by ˆ

O, is an opera-

tor able to classify the total Hilbert space Hinto two sub-

spaces. In one subspace, states have a speciﬁc property

we are interested in. These states are the Good states

and correspond, in our case, to states that respect the

symmetry S. The other states are Bad states and are

the ones that do not respect the symmetry. They belong

to the complementary subspace of HS, denoted hereafter

by H¯

A second property of the oracle is that it acts diﬀer-

ently on the good and bad states. In general, the action

is relatively simple. Here, we assume that the oracle mul-

tiplies by the same phase ϕ(resp. µ) states that do (resp.

do not) respect a symmetry, i.e.:

O|ψki=(eiϕ|ψkiif |ψki∈HS

eiµ|ψkiif |ψki∈H¯

.(2)

We consider below ϕ6=µ. Note that standard search

algorithms usually assume eiϕ =−1 and eiµ = +1. Here,

we deﬁne the oracle in a more general way because using

phases diﬀerent from the standard choice will be useful

for certain calculations shown below. The states {|ψki}

correspond here to a complete general basis that spans

the entire Hilbert space. Then, the oracle applied to a

general wave function |Ψi=Pkck|ψkigives:

O|Ψi=eiϕ X

ψk∈HS

ck|ψki

| {z }

≡|ΨGi

+eiµ X

ψk/∈HS

ck|ψki

| {z }

≡|ΨBi

.(3)

We have that |Ψi=|ΨGi+|ΨBiwhere |ΨGi(resp. |ΨBi)

corresponds to the projection of the state onto the good

(resp. bad) subspace.

|0iH H

|Ψiˆ

Oˆ

FIG. 1: Hadamard test to get the expectation value of the

operator ˆ

Uin a determined subspace deﬁned by ˆ

O(see text

and Eq. (2)). ˆ

Ostands for the oracle operator. In this circuit,

one ancillary qubit is used. The repeated measurements of 0

and 1 of the ancillary qubit give access to the real part (or

to the imaginary part if a phase gate of phase −π/2 is added

after the ﬁrst Hadamard gate on the ancilla qubit) of the

expectation value given by Eq. (4).

We then perform a Hadamard test using a generic op-

erator ˆ

Uand the oracle. The corresponding circuit is

shown in Fig 1. The choice of the operator ˆ

Uis dis-

cussed below. The Hadamard test gives access to the

following expectation value:

hΨ|ˆ

Uˆ

O|Ψi= (hΨG|+hΨB|)ˆ

Ueiϕ|ΨGi+eiµ|ΨBi,

=eiϕ hΨG|ˆ

U|ΨGi+hΨB|ˆ

U|ΨGi

+eiµ hΨG|ˆ

U|ΨBi+hΨB|ˆ

U|ΨBi.(4)

This expression simpliﬁes if ˆ

Ualso preserves the symme-

try S, so that we have ˆ

U|ΨGi ∈ HSand ˆ

U|ΨBi ∈ H¯

Accordingly, we deduce

hΨB|ˆ

U|ΨGi=hΨG|ˆ

U|ΨBi= 0.(5)

With the use of the Hadamard test, we can retrieve the

real or imaginary part of the expectation value:

hΨ|ˆ

Uˆ

O|Ψi=eiϕhΨG|ˆ

U|ΨGi+eiµhΨB|ˆ

U|ΨBi.(6)

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RestoringbrokensymmetriesusingoraclesEdgarAndresRuizGuzmanandDenisLacroixyUniversiteParis-Saclay,CNRS/IN2P3,IJCLab,91405Orsay,France(Dated:October21,2022)Wepresentanewmethodtoperformvariationafterprojectioninmany-bodysystemsonquantumcomputersthatdoesnotrequireperformingexplicitprojection.Thetechni...

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Restoring broken symmetries using oracles Edgar Andres Ruiz Guzmanand Denis Lacroixy Universit e Paris-Saclay CNRSIN2P3 IJCLab 91405 Orsay France

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