Pauli Twirling Noise Tailoring

Short answer: Pauli twirling, also called randomized compiling, inserts random single-qubit Pauli gates around noisy gates so that the ideal circuit is unchanged but the noise is converted into stochastic Pauli errors. Wallman and Emerson proposed the method in Physical Review A in 2016. IBM's documentation says it can mitigate coherent noise, which tends to accumulate quadratically with the number of operations, whereas Pauli noise accumulates linearly. It is often combined with other mitigation techniques.

The problem with coherent errors

Wallman and Emerson's abstract says a realistic quantum computer will experience errors from the environment and imperfect control, and that when these errors are even partially coherent they present a major obstacle to robust computation. Coherent errors can add up across a circuit in ways that stochastic errors do not.

What the 2016 proposal does

The paper proposes introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. The authors prove that this randomization tailors the noise into stochastic Pauli errors, and report that this can dramatically reduce error rates while introducing little or no experimental overhead. They also prove the technique is robust to variation in errors over the randomizing gates, and illustrate numerically the reductions in worst-case error. They state that with tailored noise, gates with significantly lower fidelity, comparable to fidelities realized in current experiments, are sufficient to achieve fault-tolerant quantum computation. The worst-case error rate of the tailored noise can be measured through randomized benchmarking. These are theoretical results with numerical illustration.

Later analysis

A later paper on arXiv (2212.07500) says randomized compiling tailors arbitrary Markovian errors into stochastic Pauli noise, and proves it also tailors non-Markovian errors into local stochastic Pauli noise, while investigating the technique's limitations. The authors say it alters errors in three ways. It prevents the coherent accumulation of errors, including crosstalk, across gate cycles by destroying intercycle coherent correlations. It converts individual gate cycle errors into Pauli noise. And it reduces the variability inherent to noisy devices. They report confirming these predictions on the IBM Quantum Experience platform, with data they describe as showing a drastic performance improvement across public devices.

How it is implemented in practice

IBM's documentation describes twirling as converting arbitrary noise channels into channels with more specific structure. A single circuit is replaced by a random ensemble of circuits that all have the same ideal effect, and samples are drawn from several random instances rather than one. Because most errors on current hardware come from two-qubit gates, the technique is often applied only to native two-qubit gates, such as CNOT and ECR. The documentation lists options for the number of randomizations and the shots per randomization.

What this does not tell you

Twirling shapes the noise. It does not remove it. The documentation presents it as often combined with other techniques that work better with Pauli noise. The figures in the papers apply to the devices and models tested, and the later paper explicitly studies limitations. Results on other hardware may differ.

FAQ

What is Pauli twirling?

Sandwiching gates with random Pauli gates so the ideal effect is unchanged but the noise becomes Pauli noise.

Is it the same as randomized compiling?

IBM's documentation says twirling is also known as randomized compiling, and Pauli twirling is a special kind.

Why does Pauli noise matter?

IBM's documentation says coherent noise accumulates quadratically with operations, whereas Pauli noise accumulates linearly.

Why twirl only two-qubit gates?

Most errors on current hardware come from two-qubit gates, per IBM's documentation.

Does it help with non-Markovian errors?

The 2022 arXiv paper proves randomized compiling tailors them into local stochastic Pauli noise.