# Randomized benchmarking with a tractable continuously generated group

This is a Perspective on "Matchgate benchmarking: Scalable benchmarking of a continuous family of many-qubit gates" by Jonas Helsen, Sepehr Nezami, Matthew Reagor, and Michael Walter, published in Quantum 6, 657 (2022).

By Martin Kliesch (Heinrich Heine University Düsseldorf, Germany).

## 1 Background

For any given implementation of a quantum computer one would like to have a precise description of what its logical components are. Ideally, this information would include full tomographic knowledge of the initial states, the gates and the measurements. Moreover, one would like to know how well these components act together. Obtaining such information is the central task in the characterization of quantum systems [1].

Such information on a quantum device is important for several reasons. First, it can be used to establish trust that a device functions as desired. For instance, for a device built to demonstrate quantum supremacy one needs to provide evidence that its outputs are sufficiently close to the desired ones [2]. Second, characterization methods can be used to extract information about noise in a device in order to provide a basis for iteratively improving it. Also, the software run on the device can be tailored to the noise in order to reach better performance. Finally, characterization methods can provide benchmarks for comparing different device implementations and platforms.

One challenge for the characterization of quantum devices from measurements is that one needs to deal with noise in all components. In textbook quantum process tomography this challenge is not taken into account: here one assumes that one can probe a quantum process with well-known state preparations and measurements. There are two main approaches to relax such assumptions. One is gate-set tomography [3] which aims at a self-consistent reconstruction of all noisy components from running certain types of gate sequences, which can also be drawn uniformly at random for some fixed length [4].

The other approach is randomized benchmarking (RB) and also relies on such random sequences. The idea of RB is to measure output fidelities of states generated by such sequences for different lengths [5,6,7]. Crucially, these fidelities decay in the sequence length in a way that is independent of state-preparation and measurement (SPAM) errors, up to some overall prefactor. Under certain assumptions [8] this decay curve is a mixture of (matrix) exponential decays. In particular, when the gates are drawn from a group, each exponential in the mixture corresponds to an irreducible representation of the group.

Several RB protocols have been developed for different groups with a focus on subgroups of the unitary group, see Ref. [8] for a discussion. So far the focus has been on finite subgroups. This implies that the application of the Hamiltonian generator of some group element for only a fraction of the time results in a unitary that is, in general, not in the group. Hence, common RB protocols are not compatible with fractional gates. In contrast, in many experiments fractional gates can simply be implemented by running their Hamiltonian evolution only for a fraction of the time.

Helsen et al. [9] have now formulated an RB protocol that is based on the group of so-called matchgates. This group has the nice property that fractional matchgates are again matchgates.

## 2 Matchgate RB

In order for RB to be scalable the underlying group needs to be tractable. Commonly this is achieved by choosing it to be the Clifford group or one of its subgroups. However, also non-interacting fermions are efficiently simulable [10]. In particular, this holds for any time evolution under a non-interacting Hamiltonian. Such time evolution operators are representations of the orthogonal group $O(2n)$ in the case of $n$ fermionic modes.

Fermionic systems can be mapped to qubits using a fermion-to-qubit mappings. The most popular such mapping is the Jordan-Wigner transform, which takes a free fermionic evolution, given by a transformation in $O(2n)$, to a unitary acting on $n$ qubits. Such unitaries compose the group of (generalized) matchgates. This group is also generated by all fractional Pauli $XY$, $XX$ and $YY$ unitaries acting on nearest neighbors on a chain of qubits (and an $X$ gate on the last qubit).

Helsen et al. [9] have turned these insights into an RB protocol. For this purpose, they have identified the irreducible representations of the generalized matchgate group as spaces spanned by products of $k$ Majorana operators for $k=1, \dots, n$, where $n$ is still the number of qubits. Then they use a form of character benchmarking [11,8] to separate the exponential decays from the different irreducible representations in the classical post-processing of the measurement data. This allows for the extraction of the related decay parameters, also called \emph{Majorana fidelities}. It is shown that they can provide more fine-grained information on the gate implementations than Clifford RB.

In the RB literature one often makes the assumption of gate-independent noise in order to be able to more easily analyze RB protocols. Under this assumption, measures including the average gate fidelity, a measure of trace-loss, and a measure of parity preservation can be extracted from the Majorana fidelities.

The analysis of the statistical stability of the protocol requires quite some technical effort. Already the noiseless case is technically involved: here, as a technical main result, the scaling of the variance over the random gate sequences and the measurement shot noise is proven to be polynomially bounded. This provides evidence that the Majorana fidelities can be estimated in a scalable way once the gate implementations are good enough. At the same time, the protocol is practical for actual experiments, as demonstrated on a Rigetti quantum computer.

## 3 Outlook

Matchgate RB adds a new protocol to the toolbox for the characterization of quantum devices in a way that is robust against SPAM errors. It is expected to be particularly useful for experiments. In comparison to other RB protocols it stands out in that it is based on a Lie group rather than on a finite group. Hence, it allows for the inclusion of Hamiltonian generators applied for arbitrary times. This feature opens up the possibility to bring RB closer to the analogue regime whereas it otherwise mostly applied for digital devices.

With their work [9], Helsen et al. have opened up several possibilities for further research also on the theoretical side. There are several ways how this work can be extended along the lines of ideas in Refs. [8,12]. This includes a full statistical analysis, if possible for general noise models. Moreover, the extension to short random circuits and a further interpretation of the Majorana fidelities are interesting research directions. One can further also modify the matchgate group and develop a combination of Clifford and matchgate RB by extending the gate set [9] in order to make it even more broadly applicable.

### ► References

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[12] L. Kong, A framework for randomized benchmarking over compact groups, arXiv:2111.10357 [quant-ph].
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