Randomized Benchmarking (RB) Protocols
Classification of RB protocols
Randomized Benchmarking (RB) protocols evaluate the ability of the quantum computer to perform quantum operations reliably. These tools are used to provide bounds on the error rate of quantum gates. The global idea is to use varied-length random quantum circuits with a specific gate set to quantify the average error rate of the gate set. The number of layers of gates \(l\) is varied with different values \(l \in \{l_1, l_2, ..., l_k\}\). The random circuits are built so that the measurement of the qubits yields a deterministic output bitstring. If the bitstring is measured, the computation is considered a success; otherwise, it is a failure. The average probability of success is extracted for each circuit length \(l\) and is used to fit an exponential decay function corresponding to the success probability:
\[p_\mathrm{success}(l) = A \alpha^l + B,\]where \(A\), \(B\) and \(\alpha\) are the fit parameters. The State Preparation And Measurement (SPAM) errors are absorbed in the coefficients \(A\) and \(B\). For a \(n\)-qubit state, the value of \(\alpha\) is related to the average gate infidelity:
\[\epsilon_\mathrm{r} = \frac{d-1}{d} (1- \alpha),\]which is linked to the average gate fidelity \(F_{avg}\) by the relation \(\epsilon_\mathrm{r} = 1- F_{avg}\). The value of \(\alpha\) is also linked to the average process infidelity (entanglement infidelity):
\[\epsilon_\mathrm{r} = \frac{d^2-1}{d^2} (1- \alpha),\]where \(d = 2^n\) is the size of the Hilbert space. In benchmarking studies, it is important to identify which type of infidelity is being used (see. VIII.B [1] for detailed discussion)
Following the classification done by J. Helsen et al. [2], a randomized benchmarking experiment is composed of the following key components:
- \(\mathbb{G}\): A group of gates which are applied during the RB protocol. The \(l\) gates composing the sequence are drawn from this group. This group is a subgroup of the unitary group (e.g., the Clifford group).
- \(\{l_1, l_2, ..., l_k\}\): A set of sequence lengths that denote the length of the random sequence of gates. Repeating the experiment for different values of \(l\) permits fitting the exponential decay function.
- \(\phi_r\): A reference implementation \(\phi_r\) mapping each gate \(g \in \mathbb{G}\) to its ideal implementation. The decomposition of this single gate to the quantum computer’s elementary gates must be specified.
- \(\nu\): A probability distribution (or a set of probability distribution) \(\nu\) defining how each gate \(g\) is selected from the group \(\mathbb{G}\). In most cases, this distribution is uniform.
- \(g_{end}\): An ending gate. It is usually used to revert the action of the sequence of the \(l\) gates over the initial state. In this case, the success bitstring is the initial state.
- \(\rho_0\): An input state. The input state is often just the pure state \(\ket{000...0}\).
- \(\{\Pi_i\}_{i \in \mathcal{I}}\): A Positive Operator Value Measure (generalization of the projective measurement).
A generic quantum circuit can be drawn from these specifications with a unitary \(\rho_0\) that prepares the initial state, followed by the \(l\) sequences of gates, the reversing gate \(R\), and the POVM:
The fitting of the exponential function is based on the assumption that:
- The noise is both time and gate independent
- The noise is Markovian (assumption done in almost all RB protocols), i.e., the noise does not depend on previous gates operations.
List of RB protocols
- Clifford Randomized Benchmarking
- Other gate set Randomized Benchmarking
- Randomized Benchmarking for cross talk errors
- Randomized Benchmarking for individual gate error rate assessment
- Other protocols based on Randomized Benchmarking
Extra references
The reader can refer to the references [1], [2] for a high-level overview over the different RB protocols. The reader may also refer to a gentle introduction on RB in [5].
References
- [1]A. Hashim et al., “A Practical Introduction to Benchmarking and Characterization of Quantum Computers.” arXiv, 2024 [Online]. Available at: https://arxiv.org/abs/2408.12064
- [2]J. Helsen, I. Roth, E. Onorati, A. H. Werner, and J. Eisert, “General Framework for Randomized Benchmarking,” PRX Quantum, vol. 3, no. 2, p. 020357, Jun. 2022, doi: 10.1103/PRXQuantum.3.020357. [Online]. Available at: https://link.aps.org/doi/10.1103/PRXQuantum.3.020357
- [3]J. Emerson, R. Alicki, and K. Życzkowski, “Scalable noise estimation with random unitary operators,” Journal of Optics B: Quantum and Semiclassical Optics, vol. 7, no. 10, p. S347, 2005.
- [4]C. Dankert, R. Cleve, J. Emerson, and E. Livine, “Exact and approximate unitary 2-designs and their application to fidelity estimation,” Physical Review A—Atomic, Molecular, and Optical Physics, vol. 80, no. 1, p. 012304, 2009.
- [5]A. Silva and E. Greplova, “Hands-on Introduction to Randomized Benchmarking,” arXiv preprint arXiv:2410.08683, 2024.