Study of noise in virtual distillation circuits for quantum error mitigation

2025-05-02 0 0 1.48MB 14 页 10玖币

Study of noise in virtual distillation circuits for quantum error

mitigation

Pontus Vikstål1, Giulia Ferrini1, and Shruti Puri2,3

1Wallenberg Centre for Quantum Technology, Department of Microtechnology and Nanoscience, Chalmers University of Technology, 412

96 Gothenburg, Sweden

2Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA

3Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA

Virtual distillation has been proposed as an

error mitigation protocol for estimating the ex-

pectation values of observables in quantum al-

gorithms. It proceeds by creating a cyclic per-

mutation of Mnoisy copies of a quantum state

using a sequence of controlled-swap gates. If

the noise does not shift the dominant eigen-

vector of the density operator away from the

ideal state, then the error in expectation-value

estimation can be exponentially reduced with

M. In practice, subsequent error mitigation

techniques are required to suppress the eﬀect

of noise in the cyclic permutation circuit it-

self, leading to increased experimental com-

plexity. Here, we perform a careful analysis

of the eﬀect of uncorrelated, identical noise

in the cyclic permutation circuit and ﬁnd that

the estimation of expectation value of observ-

ables are robust against dephasing noise. We

support the analytical result with numerical

simulations and ﬁnd that 67% of errors are

reduced for M= 2, with physical dephasing

error probabilities as high as 10%. Our re-

sults imply that a broad class of quantum al-

gorithms can be implemented with higher ac-

curacy in the near-term with qubit platforms

where non-dephasing errors are suppressed,

such as superconducting bosonic qubits and

Rydberg atoms.

1 Introduction

Fault-tolerant quantum error correction is necessary

for scalable quantum computation [1], however the as-

sociated hardware-performance requirements and re-

source overheads are hard to meet with the noisy

intermediate-scale quantum processors available to-

day. Consequently, for near-term applications alter-

native techniques to mitigate the eﬀect of noise have

been developed. Some of these techniques are based

on scaling noise [2,3,4,5] or learning about the ef-

fect of noise to predict the noise-free behavior of the

Pontus Vikstål: e-mail: vikstal@chalmers.se

quantum protocol [6,7], while others exploit the sym-

metry properties of the noise-free quantum circuit to

ﬂag errors [8,9,10,11,12]. Algorithm- and noise-

speciﬁc error mitigation techniques have also been

proposed [13,14].

Recently an error mitigation scheme known as vir-

tual distillation, or error suppression by derangement,

has been shown to achieve an exponential suppression

of errors in the estimation of the expectation value of

an observable [15,16,17]. The key idea behind this

protocol is to compute the expectation value of an

observable by performing measurements on a cyclic-

permutation of Mcopies of a noisy quantum state. If

the eﬀect of noise is to mix the ideal noise-free state

with orthogonal error states, then symmetries of the

cyclic-permutation state suppress the contribution to

the expectation value from the error states exponen-

tially in M.

The most straightforward approach to virtual distil-

lation is to prepare the cyclic-permutation state using

an auxiliary qubit and controlled-SWAP (CSWAP)

gates. In practice, this circuit will be prone to er-

rors, limiting the accuracy of expectation-value esti-

mation without resorting to further noise mitigation

techniques, like zero-noise extrapolation [2,3,16].

However, zero-noise extrapolation not only adds to

the sampling cost, but also considerably increases the

circuit complexity as it requires the ability to scale

the noise strength in the quantum circuit either by

scaling gate times or by adding more gates into the

circuit [4,5,18,19,20]. Thus, in this paper we

further investigate the eﬀect of noise in the virtual

distillation circuit and determine analytically condi-

tions under which its faults may be less detrimen-

tal, obviating the need for additional error mitigation.

We corroborate our ﬁndings with numerical simula-

tions of the Quantum Approximate Optimization Al-

gorithm (QAOA). Noise in virtual distillation circuits

was previously considered numerically in the context

of Heisenberg quench [17] as well as for the variational

quantum eigensolver [21], displaying robustness of the

error mitigation procedure.

We consider three commonly studied types of

noise: depolarizing, dephasing, and amplitude damp-

Accepted in Quantum 2024-06-25, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.15317v2 [quant-ph] 10 Jul 2024

ing noise and ﬁnd that the mitigated expectation

value with virtual distillation is robust against de-

phasing noise for an arbitrary even number of copies

M. We support our analysis with numerical simula-

tion of QAOA for solving a MaxCut problem of par-

titioning the set of vertices in a given graph into two

subsets such that the number of edges shared between

the two partitions is maximized, for the case of two

copies, M= 2. QAOA is implemented by preparing a

variational quantum state using a short-depth quan-

tum circuit and estimating its energy, i.e. the expec-

tation value of the Ising-Hamiltonian associated with

the MaxCut problem. The parameters in the circuit

are varied until a minimum in the energy landscape

is found. In order to overcome the adverse eﬀects of

noise in ﬁnding a state that minimizes the energy, we

combine the QAOA protocol with virtual distillation.

We ﬁnd that the error in estimating the energy with

virtual-distillation with M= 2 is reduced by 67%

when the underlying source of noise is single-qubit

pure dephasing errors at rate of 10%, compared to

20% when the underlying source of noise is single-

qubit depolarizing errors at the same rate. Addi-

tionally, we found that amplitude damping errors was

detrimental to the virtual distillation circuit, resulting

in no error mitigation. Our ﬁndings imply that vir-

tual distillation in a system in which non-dephasing

errors are suppressed compared to dephasing errors is

successful at reducing errors in expectation-value es-

timation of observables diagonal in the computational

basis without additional error mitigation schemes. It

is known that such an error channel is relevant for

Kerr-cat qubits in superconducting microwave cir-

cuits [22,23] and Rydberg atomic qubits [24] not only

when the qubits are idle but also during implementa-

tion of Toﬀoli and controlled-not gates. These two

gates can be combined to implement a CSWAP [25]

and thus it is possible to realize robust virtual distil-

lation in these platforms.

This paper is organized as follows: In Section 2 we

review the virtual distillation protocol. We analyze

the eﬀect of noise in the virtual distillation circuit

on the estimated expectation value in Section 3. We

support our analysis with numerical simulations in

Section 4 and ﬁnally, we give our concluding remarks

in Section 5.

2 Virtual Distillation

We begin this section by establishing the nota-

tion used throughout this paper, which is based on

Ref. [17]. A boldfaced superscript, for example Oi,

will be used to indicate that the operator Oacts on

the ith subsystem. We use superscript with parenthe-

ses to indicate an operator acting on multiple subsys-

tems. For instance, S(M)indicates that Sacts on M

subsystems.

Consider the output density operator, ρ, of an N-

qubit noisy quantum circuit with the spectral decom-

position

ρ=d

k=1

λk|ψk⟩⟨ψk|.(1)

Here d= 2Nand λkis the probability that the sys-

tem is found in the state |ψk⟩when measuring in the

eigenbasis of ρ. We assume, for convenience, that

the probabilities λkare listed in descending order

λ1> λ2. . . > λd. In the virtual distillation protocol,

raising ρto the power of Mand normalizing it results

in a density operator that approaches the dominant

eigenvector |ψ1⟩⟨ψ1|exponentially fast with M, i.e.

˜ρ=ρM

Tr(ρM)=Pd

k=1 λM

k|ψk⟩⟨ψk|

k=1 λM

.(2)

In virtual distillation, the expectation value of an ob-

servable Ois estimated with respect to the exponen-

tiated density matrix ˜ρ,

⟨O⟩mitigated := Tr(O˜ρ) = TrOρM

Tr(ρM).(3)

When |ψ1⟩corresponds to the output of the ideal

(noise free) quantum circuit, then ⟨O⟩mitigated ap-

proaches the ideal expectation value exponentially

fast with M. This condition is satisﬁed when noise

in the quantum circuit maps the ideal states to states

that are orthogonal to it, otherwise the dominant

eigenvector will drift and limit the error suppression

eﬃciency [16,17]. In general, for a multi-qubit state,

single-qubit errors can cause drift of the dominant

eigenvector. However, in real-world applications, this

drift is expected to be small, as also validated by the

numerical simulations in this paper. Furthermore, the

severity of this drift, or coherent mismatch, is ex-

ponentially smaller than the incoherent decay of ﬁ-

delity [26].

Note that, in virtual distillation, ⟨O⟩mitigated is cal-

culated without explicitly preparing the state ˜ρ, hence

the name “virtual”. Instead, virtual distillation uses

Mcopies of the state ρtogether with collective mea-

surements that only allow symmetric states of the

form |ψ⟩⊗|ψ⟩. . . |ψ⟩to contribute to the expecta-

tion value of O. More speciﬁcally, in Ref. [17] it was

shown that Eq. (3)is equivalent to

⟨O⟩mitigated := TrO(M)S(M)ρ⊗M

TrS(M)ρ⊗M,(4)

where O(M)is the symmetrized version of the opera-

tor O,

O(M)=1

i=1

Oi,(5)

and S(M)is the cyclic shift operator that act on all

Msubsystems. Its eﬀect is only to let symmetric

Accepted in Quantum 2024-06-25, click title to verify. Published under CC-BY 4.0. 2

states of ρ⊗Mto contribute to the expectation value

of Eq. (4),

S(M)|ψ1⟩⊗|ψ2⟩. . . |ψM⟩=|ψ2⟩⊗|ψ3⟩. . . |ψ1⟩.(6)

To measure the observable S(M)in Eq. (4)virtual

distillation uses a procedure similar to the Hadamard

test [27]. The procedure begins by preparing Mcol-

lective copies of the state ρtogether with an auxil-

iary qubit in the state |+⟩= (|0⟩+|1⟩)/√2. Next,

a sequence of CSWAP gates applies S(M)to the M

copies of ρconditioned on the auxiliary qubit being

in state |1⟩. Finally the auxiliary qubit is measured

in the X-basis and its expectation value equals to

TrS(M)ρ⊗M, i.e. the denominator of Eq. (4). Since

O(M)commutes with S(M), these two operators can

be simultaneously diagonalized, allowing them to be

measured at the same time. By also measuring O(M)

on the Msubsystems ρ, the measurement outcome

can be used together with the measurement outcome

from the auxiliary qubit to estimate the numerator

TrO(M)S(M)ρ⊗M.

3 Noise in virtual distillation circuits

We model a noisy gate in the virtual distillation cir-

cuit as an ideal gate followed by independent and

identical single-qubit errors acting on each qubit par-

ticipating in the gate. We examine three types of

single-qubit noise channels. The ﬁrst one is the de-

polarizing channel which describes a process where

information is completely lost with some probability

ϵ, and is given by [28]

Λdep(ρ) = (1 −ϵ)ρ+ϵ

3(XρX +Y ρY +ZρZ),(7)

where {X, Y, Z}are the Pauli operators and ϵis

the error probability. The second one is the pure-

dephasing channel which is a biased noise channel1and

describes loss of phase information with a probability

ϵ,ΛZ(ρ) = (1 −ϵ)ρ+ϵZρZ. (8)

The third and ﬁnal channel that we consider is the

amplitude damping channel which is characterized by

energy dissipation to the ground state over time. Al-

though no analytical expression of the mitigated ex-

pectation value for the amplitude damping channel is

derived in this work, it is deﬁned as follows:

Λamp(ρ) = K1ρK†

1+K2ρK†

2.(9)

where the Kraus operators K1and K2are given by

K1=1 0

0√1−γ, K2=0√γ

0 0 .(10)

1Of course we could have chosen an error channel with bi-

ased X- or Y-noise but we can always redeﬁne the computa-

tional basis states on Bloch sphere and call all of these Z-biased

noise.

with

γ≡4(√1−ϵ+ϵ−1).(11)

We use these deﬁnitions of the error channels because

their average channel ﬁdelities are the same for a given

ϵ, allowing for a consistent comparison across the dif-

ferent noise models.

In the next section we will present analytical results

on how the depolarizing and dephasing noise channels

aﬀect the mitigated expectation value of virtual dis-

tillation as well as their associated variances. For the

amplitude damping channel, instead, corresponding

analytical expressions could not be obtained, and nu-

merical results on that channel will be presented in

the later Section 4.

3.1 Noisy mitigated expectation values

In this section we provide the main results of this pa-

per. We derive an expression for the noisy mitigated

expectation value for even number of copies. For any

number of copies M, the cyclic shift operator S(M)

factorizes into a tensor product of M N/2number of

SWAPs, and its controlled version factorizes into a

product of MN/2CSWAP-gates. For the virtual dis-

tillation circuit, we assume that a single-qubit noise

channel is applied after each gate to the qubits in-

volved, see FIG. 1. For even number of copies M,

only one swap per subsystem is required, as for ex-

ample shown for the case of M= 4 in FIG. 1b. As a

consequence, for the case of even number of copies we

ﬁnd the following analytical expression for the miti-

gated expectation value:

⟨O⟩Λ

mitigated =Tr¯

Λ(O)ρM

Tr(ρM),(12)

where ¯

Λ = Λ⊗. . .⊗Λis a tensor product of Nsingle-

qubit error channels and Λ∈ {Λdep,ΛZ}. The details

of the calculations are provided in Appendix A. For

the case of an odd number of copies, the calculation is

more involved. Consider for instance the case of three

copies. In this case, one of the copies needs to be

swapped twice, making the mathematical derivation

of the mitigated expectation value signiﬁcantly more

diﬃcult. From Eq. (12)we see that the inﬂuence of

errors on the mitigated expectation value will depend

on the observable O. Since a general observable on

Nqubits can be expressed as a sum of N-qubit Pauli

strings from the set {I, X, Y, Z}⊗N, it is suﬃcient to

consider O∈ {I, X, Y, Z}⊗N. In this case we ﬁnd that

the mitigated expectation value for the two types of

noise are given by

⟨O⟩Λdep

mitigated =1−4

3ϵkTrOρM

Tr(ρM),(13)

⟨O⟩ΛZ

mitigated = (1 −2ϵ)k′TrOρM

Tr(ρM),(14)

Accepted in Quantum 2024-06-25, click title to verify. Published under CC-BY 4.0. 3

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StudyofnoiseinvirtualdistillationcircuitsforquantumerrormitigationPontusVikstål1,GiuliaFerrini1,andShrutiPuri2,31WallenbergCentreforQuantumTechnology,DepartmentofMicrotechnologyandNanoscience,ChalmersUniversityofTechnology,41296Gothenburg,Sweden2DepartmentofAppliedPhysics,YaleUniversity,NewHaven,Con...

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Study of noise in virtual distillation circuits for quantum error mitigation

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