Efficient Mean-Field Simulation of Quantum Circuits Inspired by Density Functional Theory

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Eﬃcient Mean-Field Simulation of Quantum

Circuits Inspired by Density Functional Theory

Marco Bernardi∗,†

†Department of Applied Physics and Materials Science, California Institute of Technology,

Pasadena, CA 91125, USA.

‡Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA.

E-mail: bmarco@caltech.edu

Exact simulations of quantum circuits (QCs) are currently limited to ∼50 qubits be-

cause the memory and computational cost required to store the QC wave function scale

exponentially with qubit number. Therefore, developing eﬃcient schemes for approxi-

mate QC simulations is a current research focus. Here we show simulations of QCs with

a method inspired by density functional theory (DFT), a widely used approach to study

many-electron systems. Our calculations can predict marginal single-qubit probabili-

ties (SQPs) with over 90% accuracy in several classes of QCs with universal gate sets,

using memory and computational resources linear in qubit number despite the formal

exponential cost of the SQPs. This is achieved by developing a mean-ﬁeld description

of QCs and formulating optimal single- and two-qubit gate functionals −analogs of

exchange-correlation functionals in DFT −to evolve the SQPs without computing the

QC wave function. Current limitations and future extensions of this formalism are

discussed.

arXiv:2210.16465v3 [quant-ph] 19 Oct 2023

1. Introduction

Noisy intermediate-scale quantum devices promise exciting advances in quantum algorithms

with no classical counterpart1–3. Classical simulations remain essential to understand the

physics of these quantum devices, improve their design, and accelerate their progress4–7.

An important direction is the development of approximate schemes that are both accurate

and computationally eﬃcient, enabling simulations of generic QCs with arbitrary depth and

degree of entanglement, ideally with favorable computational scaling. Work in this area has

focused on tensor network matrix product states to simulate QCs with a range of structures,

gate types, entanglement, and noise8–14, and more recently on simulations of generic QCs

using neural-network quantum states15. Despite these notable advances, approximate QC

simulations remain an area of active investigation.

There is an intriguing parallel between many-electron and many-qubit systems. In the

many-electron problem −a grand challenge in chemistry and materials physics16 −exact

solutions are possible only for systems with one electron (the hydrogen atom). Therefore,

unlike QC simulations, electronic structure calculations of molecules and materials are dom-

inated by approximate methods16–23, among which density functional theory (DFT) is the

main workhorse. Leveraging a mean-ﬁeld description centered on the electron density, DFT

achieves low-polynomial scaling with system size, enabling studies of matter with thousands

of interacting electrons17,24. Methods to study QCs with a similar trade-oﬀ of cost and ac-

curacy would be expedient. Early work on relating DFT to QCs focused on formal mapping

of QCs onto lattice fermions25 or connecting time-dependent DFT and spin Hamiltonians26.

These notable eﬀorts diﬀer in method and scope from this work.

Here we show a DFT-inspired approach for QCs −in short, QC-DFT −able to accurately

simulate single-qubit probabilities (SQPs) in QCs with computational cost scaling linearly

with qubit number and depth, despite the formal exponential cost of the SQPs. We present

results for various random QCs using two diﬀerent universal gate sets, and demonstrate the

formulation and optimization of QC-DFT gate functionals. We also apply this formalism

to nonrandom QCs, studying how the SQP distribution changes with QC size, as well as

simulate a simple model Hamiltonian and a quantum algorithm. These results show that

even though the exact QC wave function is exponentially complex, marginal probability

distributions such as the SQPs can be obtained with a favorable trade-oﬀ of cost and accu-

racy without computing the QC wave function. Although the current formulation is limited

to QCs with low entanglement, we discuss an extension based on reduced density matrices

which may enable further progress.

2. Theory: QC-DFT and Gate Functionals

The QC wave function for Nqubits can be expanded in the computational basis as

Ψ = X

i1i2...iN

ci1i2...iN|i1i2. . . iN⟩=

2N−1

cx|x⟩(1)

where in= 0,1are basis states for a single qubit, xare binary numbers from 0 to 2N−1,

cxare state-vector amplitudes, namely expansion coeﬃcients of the QC wave function, and

|x⟩=|i1i2. . . iN⟩are N-qubit states in the computational basis (N-bit long bitstrings).

For Nqubits, accessing this wave function requires storing and manipulating 2Ncomplex

numbers, which is out of reach for modern computers for N > 50. (A laptop can handle

N≈25 qubits, and a small computer cluster N≈30 on a single core; parallelization

is needed beyond N= 30.) In a gate-based QC, the wave function evolves at each cycle

(or step) via a unitary transformation, and it can be computed exactly with a classical

algorithm by applying single- and two-qubit gates as 2×2unitary matrices and updating

pairs of amplitudes in place4,5. From the exact wave function at step s, one can obtain the

N-qubit probability distribution ˜

Ps(x) = |⟨x|Ψs⟩|2, which can be measured experimentally

but is exponentially hard to compute3.

Here we take a diﬀerent approach and focus on the evolution of each individual qubit as a

result of mean-ﬁeld interactions with single- and two-qubit gates. We deﬁne the single-qubit

probability (SQP) for qubit n, with values between 0 and 1, as the probability of measuring

qubit nin the excited state |1⟩at step s, regardless of the state of the other qubits:

p(n)

s=X

{iq, q̸=n}|⟨i1, i2, . . . , in=1, . . . , iN|Ψs⟩|2.(2)

The exact SQPs are marginals of the N-qubit probability distribution ˜

Ps(x), and are also

exponentially hard to compute because they require knowledge of the QC wave function. We

deﬁne the SQP vector at step s,ps= (p(1), p(2), . . . , p(N))s, as the set of SQPs for all qubits

in the QC. Note that the SQP vector has Ncomponents, and thus it can be stored with

memory resources linear in qubit number N. Experimentally, the SQPs can be accessed by

measuring the state of each single qubit.

We model the evolution of the SQP vector psunder the eﬀect of single- and two-qubit

gates, using an approximate mean-ﬁeld approach inspired by DFT. In a general QC, single-

and two-qubit gates are applied to a set of qubits at each step s. As a result, the SQP vector

evolves to a new value at step s+ 1:

ps+1 =fG(ps)(3)

where we deﬁne the map fGas the exact gate functional. Here we derive approximate gate

functionals which evolve independently the SQPs of qubits acted on by single-qubit gates,

and couple qubits acted on by two-qubit gates (here, CZ and CNOT). Analogous to DFT,

where the electron interactions depend on the density, here we derive qubit-gate interactions

that depend only on the SQPs, and use them to evolve the SQP vector. Recall that pis the

probability of measuring a single qubit in state |1⟩. We deﬁne a single-qubit mean-ﬁeld state

consistent with this SQP:

|p±⟩ =p1−p|0⟩ ±√p|1⟩(4)

where we use ±to take into account two opposite phases between the |0⟩and |1⟩states.

For single-qubit gates, we apply the gate Uto this mean-ﬁeld state, and then compute

the probability of measuring |1⟩while taking the phase average over the ±states. This

approach provides explicit rules to update the SQPs at each step:

ps+1 =1

±|⟨1|U|ps±⟩|2.(5)

This equation deﬁnes the local-probability approximation (LPA) gate functional. Using eq 5,

we derive the following LPA update rules for common single-qubit gates:

Pauli X and Y: ps+1 = 1 −ps

Pauli Z, S and T: ps+1 =ps(6)

H, √Xand √Y:ps+1 = 0.5.

These results show that in our mean-ﬁeld approach the Pauli X and Y gates ﬂip the SQP,

the Pauli Z, S and T gates leave the SQP unchanged as they act only on the phase, and the

Hadamard, Pauli √Xand √Ygates set the SQP to 1/2.

For the two-qubit gates considered here, CZ and CNOT, we use our intuition combined

with the LPA rules to approximate the SQP evolution. In our probability-based formulation,

the controlled unitary acts on the target qubit when p(c)>0.5, namely when the control

qubit is “more one than zero”. The probability p(c)of the control qubit is left unchanged and

the probability p(t)of the target qubit is evolved according to the respective gate:

CZ: p(t)

s+1 =p(t)

CNOT: if p(c)<0.5,p(t)

s+1 =p(t)

s(7)

if p(c)>0.5,p(t)

s+1 = 1 −p(t)

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摘要：

EfficientMean-FieldSimulationofQuantumCircuitsInspiredbyDensityFunctionalTheoryMarcoBernardi∗,††DepartmentofAppliedPhysicsandMaterialsScience,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA.‡DepartmentofPhysics,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA.E-mail:bmarco@caltech.eduExacts...

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