Benchmarking multi-qubit gates - I Metrological aspects Bharath Hebbe Madhusudhana 1Fakult at f ur Physik Ludwig-Maximilians-Universit at M unchen Schellingstrae 4 80799 M unchen Germany

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Benchmarking multi-qubit gates - I: Metrological aspects

Bharath Hebbe Madhusudhana

1Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, Schellingstraße 4, 80799 M¨unchen, Germany

2Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 M¨unchen, Germany

3Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

Accurate and precise control of large quantum systems is paramount to achieve practical advan-

tages on quantum devices. Therefore, benchmarking the hardware errors in quantum computers has

drawn signiﬁcant attention lately. Existing benchmarks for digital quantum computers involve aver-

aging the global ﬁdelity over a large set of quantum circuits and are therefore unsuitable for speciﬁc

many-qubit control operations used in analog quantum operations. Moreover, average global ﬁdelity

is not the optimal ﬁgure-of-merit for some of the applications speciﬁc to analog devices, such as the

study of many-body physics, which often use local observables. In this two-part paper,we develop a

new ﬁgure-of-merit suitable for analog/multi-qubit quantum operations based on the reduced Choi

matrix of the operation. In the ﬁrst part, we develop an eﬃcient, scalable protocol to completely

characterize the reduced Choi matrix. We identify two sources of sampling errors in measurements

of the reduced Choi matrix and we show that there are fundamental limits to the rate of conver-

gence of the sampling errors, analogous to the standard quantum limit and Heisenberg limit. A

slow convergence rate of sampling errors would mean that we need a large number of experimental

shots. We develop protocols using quantum information scrambling, which has been observed in

disordered systems for e.g., to speed up the rate of convergence of the sampling error at state prepa-

ration Moreover, we develop protocols using squeezed and entangled initial states to enhance the

convergence rate of the sampling error at measurement, which results in a metrologically enhanced

reduced process tomography protocol.

I. INTRODUCTION

A principle challenge in developing quantum hardware

is to reliably actualize a wide range of control opera-

tions on quantum systems, e.g., a set of qubits. This

is an important challenge not only in the development of

quantum computers and quantum simulators, but also in

the development of other, non-computational quantum

technologies such as quantum metrology and quantum

communications. Consequently, benchmarking quantum

operations has been a vibrant area of research lately.

There are two fundamental challenges in benchmark-

ing quantum operations. First, a quantum operation is

characterized by an exponentially large number of vari-

ables [1] and therefore a process tomography, i.e., a com-

plete characterization, is not scalable. And second, the

expected eﬀect of a quantum operation on the quantum

system is sometimes outside the classical computational

limits and desirably so, making it impossible to have a

reference to compare the experimental implementation of

the operation with. The former is a quantum metrology

problem — a solution would involve designing a proto-

col which can be implemented without adding signiﬁcant

noise, in order to measure the desired parameters of the

operation. The latter is a computational challenge — a

solution to it would involve ﬁnding eﬃciently veriﬁable

properties of the operation.

Some of the above challenges are addressed by random-

ized benchmarking protocol [2,3], by considering cyclic

quantum circuits. That is, quantum circuits that corre-

spond to a unitary evolution of . Ideally, such circuits

map each state to itself. Therefore, one can start with

an N−qubit state |0iN∈ H⊗N, which can be prepared

Figure 1. Reduced process: a. The subsystems of a sys-

tem of Nqubits can be arranged in a hierarchy of inclu-

sion. Corresponding to a state of the system, for every subset

S⊂ {1,2,3,· · · , N }, one can deﬁne a reduced density ma-

trix. These reduced density matrices form a Hierarchy under

partial trace. The various levels of the Hierarchy are charac-

terized by number of qubits in the subsystem. Figure shows a

schematic of the hierarchy for N= 3. b. Corresponding to a

process Φ acting on the full system, one can deﬁne, for every

subsystem S, a reduced process ΦS. The reduced processes

also form a Hierarchy. c. shows a tomography of the reduced

process. The measurement involves preparing a mixed initial

state, resulting in two sources of sampling errors — one at

state preparation and one at measurement.

arXiv:2210.04330v2 [quant-ph] 17 Jan 2023

Figure 2. Metrologically enhanced reduced process tomography protocol: We consider a 2D array of N×νqubits

—νstatistical repetitions of a system of Nqubits realized in parallel so that one can use metrological techniques to enhance

the readout. The protocol is designed for a reduced process tomography ΦSof a subsystem Sof the Nqubits, corresponding

to an N-qubit operation Φ. It consists of 4 steps. In Step 1, we apply entangling operations to ¯

Swithin each realization in

order to scramble the information. This reduces the sampling error in state preparation ∆2

prepΦS(see Sec. V). This step also

includes entangling gate on qubits in Sacross realizations. These are squeezing operations aimed at reducing the sampling

error in readout ∆2

measΦS(see Sec. VI). Step 2 consists of independent random single qubit gates on ¯

Sin order to prepare a

uniform mixed state. Step 3 consists of evolution under the target operation Φ and step 4 involves a readout.

reliably, one can measure the ﬁdelity of the ﬁnal state

with |0iN.His the Hilbert space of one qubit. Ideally,

one must average this ﬁdelity over various initial states.

However, arbitrary initial states cannot be prepared re-

liably on current devices. Assuming that the errors are

independent of the gates applied, we can average over

cyclic circuits instead [4]. The simplest way to produce

a cyclic circuit is to mirror a circuit — applying a set

of gates and inverting them. However, the typical depth

of the circuit should be exponential in the system size,

in order to eﬀectively represent a Haar random unitary

[5]. Moreover, this method overlooks some systematics,

which maybe erased by the mirroring. A popular alter-

native is to use Cliﬀord gates. That is, restricting the

gates used in the circuits to Cliﬀord gates [6,7]. Cir-

cuits composed of Cliﬀord gates are classically simula-

ble. Moreover, the number of gates necessary to produce

a typical Cliﬀord gate is only polynomial. However, it

is unclear whether a Cliﬀord gate benchmark provides a

useful estimate of the average ﬁdelity, given that a typical

unitary gate consists of many more gates than a typical

Cliﬀord operation. This idea has been extended to other

groups, besides the Cliﬀord group as well [8]. Recently,

there have been works combining the idea of circuit mir-

roring and Cliﬀord gates [9] to tailor the benchmarking

scheme to target errors of a speciﬁc nature. Alterna-

tively, one can consider averaging over a subgroup of

SU (2N) [10]. Recently, a new protocol to benchmark

aspeciﬁc unitary Uas opposed to averaging over several

circuits, was proposed based on the symmetries of U[11].

Another approach to benchmarking a speciﬁc unitary is

to use random matrix theory and test the expected sta-

tistical properties such as moments of the output distri-

bution. For instance, a new benchmarking method based

on emergent Porter-Thomas distributions in the output

state after time evolution under a speciﬁc many-body

Hamiltonian was recently developed [12,13]. While this

method is suitable for speciﬁc many-qubit operations, it

does not scale eﬃciently with the system size.

Here, we develop a new ﬁgure-of-merit and a proto-

col to benchmark the experimental implementation of a

given unitary U∈SU(2N), which is produced either by

a circuit Cconsisting of one and two qubit gates or by

time evolution under a many-qubit Hamiltonian H. The

former is relevant for digital quantum computers built

using ion traps/superconducting circuits/neutral atoms

and Uwould be the ordered product of the unitaries cor-

responding to the gates in C. The latter is relevant for

analog quantum simulators built for e.g., using trapped

neutral atoms and U=e−iHt where tis the duration of

the time evolution. We focus on this case in this work.

This is a desirable goal for applications such as quan-

tum certiﬁed approximations, where one uses an analog

quantum computer to benchmark the performance of a

new classical approximation ansatz, shown in the recent

work [14]. In fact, one can advance this idea further

— train a classical neural network on the data from a

quantum simulator in order to develop implicit classical

approximations [15,16]. These applications only need a

few accurate many-qubit operations. Moreover, quantum

computers based alternate gate-sets that include direct

application of many-qubit operations have been studied

recently [17,18]. One such example is a digital-analog

quantum computer, where universal control is achieved

using a combination of a few many-qubit operations

and single-qubit gates [19,20]. Experimental platforms

are also being developed [21] where, our present goal of

benchmarking a speciﬁc many-qubit operation would be

very relevant.

II. RESULTS

Every benchmarking protocol is anchored to a ﬁgure-

of-merit — the quantity which characterizes the quality

of the quantum operation and which we intend to mea-

sure through the protocol. The ﬁgure-of-merit is chosen

carefully, tailored to a desired application of the quan-

tum device. The existing benchmarking protocols use the

global ﬁdelity, averaged over a large set of circuits as the

ﬁgure-of-merit and therefore evaluate the entire device as

a whole, as opposed to what we need — a ﬁgure-of-merit

that evaluates the accuracy of implementation of the spe-

ciﬁc unitary U. Moreover, the the global ﬁdelity is not

always the relevant measure. An N−qubit state contains

a large volume of quantum information, with an intricate

structure. One can organise this information in a hierar-

chy, where the lowest strata consists of reduced density

matrices for each qubit and the higher strata consist of

correlations of various orders among subsets of the N

qubits (Fig. 1a). While the global ﬁdelity between two

quantum states represents an aggregation of the errors

incurred at various strata of the hierarchy of quantum

information, it is one number, making it hard to extract

the component of the error we may be speciﬁcally tar-

geting. One can deﬁne ﬁdelities of reduced density ma-

trices of various subsets. These ﬁdelities represent errors

coming form various sources. Building up an analogy

between states and quantum processes (see. Fig. 1), we

deﬁne a reduced process tomography i.e., tomography of a

process restricted to a small subset of the qubits in order

to develop a benchmarking protocol that addresses errors

speciﬁc to the chosen subset Fig. 1b. If S⊂ {1,2,··· , N}

is a subset of the system and Φ is a quantum process act-

ing on the whole system, we can deﬁne reduced process

on the subset Sas ΦS(ρS) = Tr ¯

S(Φ(ρS⊗1

2|¯

S|¯

S)). Here,

S={1,2,··· , N}−Sand |¯

S|is the number of qubits in

it.

The initial state in a reduced process tomography is

necessarily mixed and is produced using controlled sam-

ples that average to the target mixed state. Therefore,

we have two independent sources of sampling errors in

such a tomography measurement — one corresponding

to the initial state and the other corresponding to the

measurement (Fig. 1c). Therefore, the total sampling

error is given by

∆2ΦS= ∆2

prep.ΦS+ ∆2

meas.ΦS(1)

See ref. [22] for details and explanation for this expres-

sion. Most sampling errors with νuncorrelated samples

scale as ∆2

prep.ΦS∼1

ν(and ∆2

meas.ΦS∼1

ν). Thus, the

total square error also scales inversely with ν. There are

two main problems pertaining to benchmarking via re-

duced process tomography:

1. Metrological aspects: Enhancing the conver-

gence rate of the sampling errors in the tomography

measurements.

2. Computational aspects: Developing bench-

marks using the reduced process ΦS.

In this paper, we address the ﬁrst of the above two prob-

lems. We develop protocols to enhance the convergence

of the total sampling error. In section V and VI, we

develop entanglement-based protocols to speed up the

convergence of the sampling error in state preparation,

i.e., in ∆2

prep.ΦSand in measurement, i.e., in ∆2

meas.ΦS.

In section III and IV, we develop from background

material. The computational aspect will be addressed in

the part II of this paper [23].

III. THE REDUCED CHOI MATRIX

We refer to the experimental implementation of U, by

the map Φ. In order to maintain generality, we model

this operation by a the most general quantum operation,

i.e., a completely positive map. The Choi matrix corre-

sponding to this map is a 4N×4Nmatrix ρΦ, deﬁned on

the space H⊗N⊗ H⊗Nas ρΦ

ij;kl = Tr(Φ(|iihj|)|kihl|).

Here, |ii,|ji,|kiand |liare basis elements of H⊗N.

One can view ρΦas a block matrix, where the i, j−th

block is the 2N×2Nmatrix Φ(|iihj|). Moreover, If

we choose ρas an initial state and measure ˆ

Oafter the

quantum operation Φ, the expectation value is given by

Tr( ˆ

OΦ(ρ)) = Tr(ρΦˆ

O⊗ρ) (see ref. [22] for more details).

The unitary Uhas its own Choi matrix representation

ρU. For a given subset of qubits S⊂ {1,2,··· , N}, we

deﬁne the reduced Choi matrix as the partial trace

ρΦ,S = Tr ¯

SρΦ(2)

Here, ¯

S={1,2,··· , N}−S. Similarly, the reduced Choi

matrix for the unitary Uis ρU,S = Tr ¯

SρU. If Scontains

mqubits, the reduced Choi matrix is a 4m×4mmatrix

that represents the eﬀect of the time evolution on the

subset S. To obtain a precise interpretation, if ˆ

Ois an

observable acting on Sand ρSis a state of S, it follows

that

Tr(ρΦ,S ˆ

O⊗ρS) = Tr ˆ

O⊗¯

SΦρS⊗1

2N−m¯

S (3)

That is, the partial trace represents the process on S,

that Φ would induce on a given state ρSof S, with

the rest of the qubits initially in the uniformly mixed

state 1

2N−mN−m. Note that if Uis an entangling op-

eration, the partial trace ρU,S can be quite general and

non-trivial. The Choi matrix of a quantum operations

has all the properties of a quantum state in a higher di-

mensional Hilbert space and therefore, a partial trace is

a natural choice for reduction (See ref. [22] for more de-

tails).

The reduced Choi matrix would be aﬀected my most

errors that would also aﬀect the reduced density matrix

of S, after the time evolution. Moreover, for small m

(we use m= 1,2 below), it is possible to experimentally

measure the reduced Choi matrix. Therefore, it makes

a good candidate that can be used to benchmark the

time evolution. Eq. 3provides a way of measuring the

reduced Choi matrix. One can obtain Tr(ρΦ,S ˆ

O⊗ρ)

by preparing Sin the state ρSand the rest of the

qubits in the uniform mixed state and measuring ˆ

Oon

S, after the quantum operation. By varying ˆ

Oand ρ,

one can reconstruct the reduced Choi matrix ρΦ,S . We

refer to this process as the reduced process tomography.

We can then compare ρΦ,S with ρU,S to benchmark

the operation (the details on the eﬃciently veriﬁable

properties of ρU,S will be presented in part-II of this

paper ).

A. Reduced process tomography

The tomography of ρΦ,S is straightforward. For in-

stance, consider m= 1, i.e., S={1}.ρΦ,S is a 4 ×4

matrix and has 16 free parameters. One can choose

O, ρS∈ T =|0ih0|,|1ih1|,1

2(|0i+|1i)(h0|+h1|),

2(|0i+i|1i)(h0| − ih1|)(4)

. There are 16 such combinations of ρS,ˆ

O. One can

reconstruct ρΦ,S using these 16 measurements. This idea

extends to m > 1, with ρS,ˆ

O∈ T m. However, it is

practically challenging to go beyond m= 2, although

the protocol we present below is scalable in N.

The natural protocol, suggested by Eq. 3. For each

pair τ1, τ2∈ T m, the component of ρΦ,S is

ρΦ,S

τ1,τ2= Tr τ2⊗¯

SΦτ1⊗1

2N−m¯

S (5)

Therefore, we prepare the mqubits in Sin the state τ1,

the remaining N−mqubits in the uniform mixed state,

2N−mN−mand apply the operation Φ, followed by a

measurement of τ2on the mqubits in S. The expecta-

tion value of this measurement is the component ρΦ,S

τ1,τ2.

Two questions remain: what is the convergence rate of

the measurement to the expectation value, i.e., how many

experimental shots do we need? and how do we prepare

the N−mqubits in the uniform mixed state, 1

2N−m¯

The two are connected — the convergence rate also de-

pends on the error induced in the preparation of the uni-

formly mixed state. In other words, one can optimize the

preparation strategy to maximize the convergence rate.

Although, it is apparent that preparing a quantum

system in the uniform mixed state is straightforward,

it is experimentally quite challenging to ensure scalabil-

ity with N. An incoherent sum of Haar random states

quickly converges to the uniform mixed state for arbi-

trary N. However, a typical Haar random state is highly

entangled and in a real experiment, we can only reliably

prepare states with very low entanglement.

We consider a simple example to illustrate the prob-

lem. We can produce a single qubit uniformly mixed

state 1

2using a set of νexperimental shots, where in

each shot the qubit is prepared randomly in |0ior |1i

with equal probability. We refer to the state prepared

after νshots as ρreal,{1}= 1/ν Pν

i=1 |ziihzi|(the {1}in-

dicates we are in a single qubit system). Here, zi= 0,1.

The Uhlmann ﬁdelity between ρreal,{1}and 1

2is

Fρreal,{1},1

2=1+2ps(1 −s)

2(6)

Here, s=1

νPizi.shas an average value of 0 and

a standard deviation of 1

2√ν. Thus, the ﬁdelity has

an average of 1 −1/4ν. If we now extend this to N

qubits, i.e., pick the state of each qubit to be |0ior

|1iat random, for each shot ν, the prepared state is

ρreal =1

νPν

i=1 NN

j=1 |zij ihzij |, where each zij = 0,1

chosen at random. It is straihghtforward to show that

the Uhlmann ﬁdelity is

Fρreal,1

2N= ΠN

j=1

1+2psj(1 −sj)

2(7)

Here, sj=1

νPizij . If the state of the qubit are assumed

to independent random variables, the average ﬁdelity

is (1 −1/4ν)N. Note the exponential decay. Below,

we will formalize this result into a theorem and also

show a quantitative relation between the entanglement

of the states and the convergence rate. Before that, we

will address the important question as suggested by the

intriguing exponential scaling of the Uhlmann ﬁdelity:

what is the most appropriate measure of distance to use

in the convergence analysis?

B. Characterising the error in mixed state

preparations

The most logical way of deciding on a distance measure

to evaluate the error is using the intended purpose for

which the mixed state. is being prepared. In this case,

the purpose is to measure ρΦ,S

τ1,τ2and therefore, we will

pick a measure of the distance that is induced by the

maximum error in ρΦ,S

τ1,τ2. Looking at Eq. 3, the quantity

of interest is linear in the state of the N−mqubits in

S. Indeed, if one experimentally prepares ρreal,¯

Swhile

attempting to prepare 1

2N−m¯

S, we would obtain

˜ρΦ,S

τ1,τ2= Tr τ2⊗N−mΦτ1⊗ρreal,¯

S (8)

and the error would be

∆ρΦ,S

τ1,τ2=|ρΦ,S

τ1,τ2−˜ρΦ,S

τ1,τ2|=|Tr [τ2⊗¯

SΦ (τ1⊗)] |(9)

where =1

2N−m¯

S−ρreal,¯

S. We may rewrite this as

∆ρΦ,S

τ1,τ2=|Tr(ρΦτ2⊗¯

S⊗τ1⊗)|=|Tr(PΦ

τ1,τ2)|

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Benchmarkingmulti-qubitgates-I:MetrologicalaspectsBharathHebbeMadhusudhana1FakultatfurPhysik,Ludwig-Maximilians-UniversitatMunchen,Schellingstrae4,80799Munchen,Germany2MunichCenterforQuantumScienceandTechnology(MCQST),Schellingstr.4,80799Munchen,Germany3Max-Planck-InstitutfurQuantenoptik,Han...

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Benchmarking multi-qubit gates - I Metrological aspects Bharath Hebbe Madhusudhana 1Fakult at f ur Physik Ludwig-Maximilians-Universit at M unchen Schellingstrae 4 80799 M unchen Germany

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