Efficient characterization of qudit logical gates with gate set tomography using an error-free Virtual-Z-gate model Shuxiang Cao1Deep Lall2 3Mustafa Bakr1Giulio Campanaro1Simone D Fasciati1

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Eﬃcient characterization of qudit logical gates with gate set tomography

using an error-free Virtual-Z-gate model

Shuxiang Cao,1, ∗Deep Lall,2, 3 Mustafa Bakr,1Giulio Campanaro,1, †Simone D Fasciati,1

James Wills,1, ‡Vivek Chidambaram,1, §Boris Shteynas,1, ‡Ivan Rungger,2and Peter J Leek1, ¶

1Department of Physics, Clarendon Laboratory, University of Oxford, OX1 3PU, UK

2National Physical Laboratory, Teddington, TW11 0LW, UK

3Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, UK

Gate-set tomography (GST) characterizes the process matrix of quantum logic gates, along with

measurement and state preparation errors in quantum processors. GST typically requires extensive

data collection and signiﬁcant computational resources for model estimation. We propose a more

eﬃcient GST approach for qudits, utilizing the qudit Hadamard and virtual Z gates to construct

ﬁducials while assuming virtual Z gates are error-free. Our method reduces the computational costs

of estimating characterization results, making GST more practical at scale. We experimentally

demonstrate the applicability of this approach on a superconducting transmon qutrit.

Characterizing and modeling errors is essential for

the development of quantum processors. Understanding

these details enables the improvements of hardware com-

ponents to eliminate errors, mitigation through quantum

error mitigation strategies [1,2] and evaluating the appli-

cability of quantum error correction codes on a quantum

processor [3,4]. While these characterization methods

were initially developed for processors based on two-level

quantum systems (qubits), the recent advancement of d-

level systems (qudits) (d > 2) requires compatible tools

for characterizing the performance of qudit logic gates.

Qudit-based processors can potentially overcome techni-

cal challenges: they can reduce the number of elementary

units in a physical device [5–8], improve computational

eﬃciency [7,9–11], and simplify the implementation of

quantum gates [12,13].

Randomized benchmarking (RB) is widely used to ex-

tract average gate inﬁdelity [14,15] and has recently

been demonstrated on a superconducting qutrit [16,17].

While RB can determine the average gate ﬁdelity of

a speciﬁc gate, it does not provide detailed error in-

formation. Process tomography [18] is a protocol for

reconstructing the quantum process of a speciﬁc gate,

assuming negligible state preparation and measurement

(SPAM) errors, as well as Markovianity of the noise [19].

As an improvement on this method, gate-set tomography

(GST) [20,21] takes the SPAM errors into account; GST

performs process tomography for all gates in the gate set,

including all elementary gates for state preparation and

measurement. The quantum process can then be esti-

mated by optimizing a numerical model that includes the

process matrix and SPAM operators, using information

from the entire gate set. GST has been applied to charac-

terize quantum processes in superconducting qubits [22–

24], ion traps [4], and nuclear spin qubits [25]. It has

also been used for time-domain tracking to analyze pa-

rameter drift in quantum control [26,27]. Although the

techniques for qubit tomography are well-established, the

development of optimal methods for designing GST ex-

periments for qudits still requires further exploration.

In this letter, we propose and demonstrate an eﬃcient

GST method for the characterization of qudits. Our

method utilizes only qudit Hadamard gate and virtual

Z gates for state preparation and measurement in dif-

ferent bases. By assuming that the virtual Z gates are

ideal, we simplify the model to reduce the computational

cost of the GST estimation process. We implement this

method on a superconducting transmon qutrit, extract-

ing the full process matrices and SPAM errors to validate

its practicality. We compare the characterization results

with those from a model that fully parameterizes the vir-

tual Z gates and those assuming ideal virtual Z gates, as

well as with results from RB demonstrating its validity.

The goal of GST is to reconstruct the quantum pro-

cess of all the gates in the gate set G, taking into account

that the initial state preparation and measurements are

imperfect. The following discussion utilizes the superop-

erator formalism [28], representing the density operator

ρand measurement operator Eas a vector in Schmidt-

Hilbert space, denoted by superket |ρ⟩⟩ and superbra

⟨⟨E|, respectively [21] (see the supplementary materials

for more details). In this letter, we perform maximum

likelihood GST [20,29], which collects the following data

from the measured probability distributions:

mijkl =⟨⟨El|F(m)

iGkF(p)

j|ρ0⟩⟩,(1)

where |ρ0⟩⟩ is the initial state and ⟨⟨El|is the measure-

ment basis that can be directly implemented on the hard-

ware. The ﬁducials F(p)

jrepresent the quantum processes

for preparing the initial states, and F(m)

ifor implement-

ing the measurement basis. Sandwiched between the

ﬁducials, Gk∈ G is the quantum process in the gate

set, which contains all the gates we are interested in, and

the gates used to implement F(m)

iand F(p)

j. To make the

GST more accurate, Gkis usually replaced with a gate

sequence that ampliﬁes the error, known as a sequence of

germs [30]. The estimated quantum processes ˜

Gkfor all

arXiv:2210.04857v4 [quant-ph] 11 Jul 2024

Gkcan be found with the maximum likelihood method

by minimizing the objective function

Lm(˜

El,˜

F(m)

i,˜

Gk,˜

F(p)

j, ρ)

ijkl

(⟨⟨ ˜

El|˜

F(m)

i˜

Gk˜

F(p)

j|ρ⟩⟩ − mijkl)2,(2)

where ˜

El,˜

F(m)

i,˜

Gk,˜

F(p)

j, and ˜ρ0are the estimated val-

ues for the physical operators El,F(m)

i,Gk,F(p)

j, and

ρ0. Minimizing this objective function is an optimization

problem that is subject to speciﬁc physical constraints,

which ensure that all the estimated operators are physi-

cal [30]. Additionally, the optimization problem exhibits

a gauge freedom, represented by:

⟨⟨ ˜

El|˜

F(m)

i˜

Gk˜

F(p)

j|ρ⟩⟩

=⟨⟨ ˜

El|B(B−1˜

F(m)

iB)(B−1˜

GkB)(B−1˜

F(p)

jB)B−1|ρ⟩⟩,

(3)

where Bdenotes the gauge matrix. Gauge optimization

is essential for accurately determining the process oper-

ator, as well as the initial state and measurement opera-

tors [30].

The above-mentioned optimizations are computation-

ally expensive. Firstly, the number of free parameters

that need to be optimized is proportional to the number

of gates in the gate set used to synthesize the ﬁducials

and germs. The optimizer for ﬁtting the experimental

data, for example, the Levenberg-Marquardt Algorithm

[31] used by the PyGSTi software package, requires the

evaluation of the Jacobian matrix at each optimization

step. The size of the Jacobian matrix is proportional to

the number of free parameters and to the number of ele-

ments of mijkl. The computational cost of evaluating the

Jacobian matrix is proportional to the length of the gate

sequence [21]. Similar arguments apply to the Hessian

matrix for error bar estimation, which has a size propor-

tional to the square of the number of free parameters and

to the number of elements in mijkl [21] and hence even

poorer scaling. Furthermore, the collection of mijkl from

experiments can be time-consuming. It would therefore

be best if we could minimize the size of mijkl, shorten the

gate sequence, and reduce the number of diﬀerent gates

involved in synthesizing all ﬁducials.

From the motivations above, we propose a new con-

struction of the GST model optimized for cost, leverag-

ing the existence of virtual Z gates [32]. Unlike physical

alterations of the quantum state, virtual Z gates modify

the phase of subsequent gate pulses to implement a ro-

tational frame shift, eﬀectively applying a Z gate to the

quantum state. Previous studies have shown that vir-

tual Z gates has signiﬁcantly lower errors compared to

physical gates [32]. Therefore, assuming that all virtual-

Z gates are perfect can reduce the number of parameters

FIG. 1: (a) Quantum circuit scheme for implementing

GST. (b-c) Comparison between the traditional and our

proposed parametrization models for GST. In these

diagrams, dark-colored gates are parametrized, while

light-colored gates (virtual Z gates) are ﬁxed, reducing

the number of parameters. Uirepresents the gates of

interest to be characterized, but not involved in

constructing ﬁducials. Unlike the traditional approach,

which parametrizes multiple gates in each neighboring

two-level subspace to construct ﬁducials, our method

requires parametrization only of the qudit Hadamard

gate H.

in our model and enhance its eﬃciency. In addition, it

also simpliﬁes the process for gauge optimization. The

gauge matrix Bis no longer an arbitrary process matrix;

it now has to commute with all virtual Z gates, which

signiﬁcantly reduces the gauge freedom.

To further reduce the model parameters, we pro-

pose constructing the ﬁducials using qudit Cliﬀord gates,

which can be generated by the qudit Hadamard gate H

and phase gate S[33], deﬁned as follows:

d−1

j=0

ωj(j+1)/2|j⟩⟨j|,

H=1

√d

d−1

j=0

d−1

k=0

ωjk |j⟩⟨k|,

(4)

where dis the dimension of the qudit, and ωdenotes a d-

th root of unity, i.e., ωd= 1. Note that the Sgate purely

modiﬁes the phase of the state and can be implemented

using virtual Z gates. Therefore, the GST model for the

full Cliﬀord group can be constructed by parameterizing

only the Hgate, reducing the complexity of the model

(see Figure 1). Our proposed model has better scaling

than the traditional approach by parameterizing gates in

FIG. 2: The number of parameters required to

parameterize the ﬁducial states, the initial state density

operator, and the measurement operators of a single

qudit, in relation to the dimension of the qudit.

each neighboring two-level subspace of the qudits, and

the comparison of the number of parameters required to

parameterize single qudit GST ﬁducials is shown in Fig-

ure 2. To determine the exact gates needed to construct

the ﬁducials, we select a set of Cliﬀord gates that pro-

duce a complete basis for preparation and measurement.

These gates are selected by iteratively adding a new Clif-

ford gate to the ﬁducial list, which provides access to new

basis that are all orthogonal to those accessible through

the already selected ﬁducials. With this conﬁguration,

we avoid constructing an overcomplete basis for tomog-

raphy, enabling us to use the minimum number of ﬁdu-

cials and achieve the smallest possible size for mijkl. The

detailed algorithms for this process can be found in the

supplementary materials.

We demonstrate the experimental implementation of

the proposed GST method on a superconducting trans-

mon qutrit. This study examines the feasibility of assum-

ing that virtual Z gates are ideal (the Static VZ model )

by comparing the results of processing the same dataset

with and without parameterizing the virtual Z gates (the

Full model ). The gates of interest are the X01(π/2)

and X12(π/2). Additionally, we implement qutrit RB

and compare its results with the outcomes of the two

GST models. The qutrit characterized in this paper is a

single superconducting transmon implemented in a 3D-

integrated coaxial circuit design [34,35]. For the hard-

ware details, please refer to the supplementary materials

and the previous study [36].

In this study, we choose a maximum sequence length of

512 germs, and we sample each sequence 500 times. The

ﬁducials chosen for this experiment can be found in the

supplementary materials. We show that while previous

physical proposals for tomography use an over-complete

basis with 9 measurement values [37,38], our proposal re-

quires only 4 measurement values [39]. The collected data

is processed by the PyGSTi software with our model.

Overall, the Static VZ model ﬁnishes in 56.2% of the

time it takes the fully parameterized model to complete

[40]. We expect a further increased speedup with our

FIG. 3: Comparison between GST inﬁdelity results and

the RB results. The size of the error bar indicates a

95% conﬁdence interval. The blue and red colors

represent the GST results from two diﬀerent models and

the gray color represents the RB results, respectively.

Gate Full (×103) Static VZ (×103) RB (×103)

I0.365(97) 0.328(96) N/A

Z1(2π

3) 0.221(57) N/A N/A

Z2(2π

3) 0.274(70) N/A N/A

X01(π

2) 2.112(96) 2.14(12) 1.70(44)

X12(π

2) 0.990(70) 1.72(11) 1.32(28)

H3.22(11) 3.651(12) 6.3(26)

TABLE I: Average gate inﬁdelity obtained from GST

experiments and RB experiments. The number in

brackets in this table indicates a 95% conﬁdence

interval.

method as system sizes increase, because the number of

parameters increases at a much slower rate with respect

to the size of the system compared to the traditional GST

approach.

The SPAM error is characterized by the reconstructed

initial state density matrix and the measurement oper-

ators. We found the initial state inﬁdelity 1 −F˜ρ0=

0.1137(26) and 0.0962(36) for the Full model and the

Static VZ model, respectively. The average measurement

inﬁdelity is 1 −F˜

M= 0.0348(9) and 0.0307(12) for the

Full model and the Static VZ model. For the recon-

structed operators, please see the supplementary materi-

als for more details.

We compare the inﬁdelity metric between the fully pa-

rameterized GST model, the static VZ GST model, and

the RB results (see Figure 3and Table I). We show that

the GST results for both models obtain a comparable re-

sult to the RB results for the gates of interest. However,

due to the hardware limitation of the number of gates we

could implement in the RB experiment, the error bar we

could obtain for the Hgate with RB is quite large, and

we could not draw a conclusion (see the supplementary

materials for more details).

The advantage of GST over RB is that it provides a

detailed characterization of the nature of errors. We use

error generators [41] to examine the error details. For

an ideal target map ˆ

Giand a physically reconstructed

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Efficientcharacterizationofquditlogicalgateswithgatesettomographyusinganerror-freeVirtual-Z-gatemodelShuxiangCao,1,∗DeepLall,2,3MustafaBakr,1GiulioCampanaro,1,†SimoneDFasciati,1JamesWills,1,‡VivekChidambaram,1,§BorisShteynas,1,‡IvanRungger,2andPeterJLeek1,¶1DepartmentofPhysics,ClarendonLaboratory,Un...

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Efficient characterization of qudit logical gates with gate set tomography using an error-free Virtual-Z-gate model Shuxiang Cao1Deep Lall2 3Mustafa Bakr1Giulio Campanaro1Simone D Fasciati1

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