Error Per Layered Gate Protocol (EPLG)

Motivation

This protocol was proposed by IBM in 2023 as a supplement of the quantum volume protocol [1]. The quantum volume only focuses on a specific high performing subset of qubits on the quantum chip. The Error Per Layered Gate probotol (EPLG) aims to evaluate the performance of a whole quantum computer chip instead. It also permits to rapidly extract lower bounds on 2-qubit error rates.

Protocol details

The protocol consits in selecting \(n\) qubits from the quantum chip used to implement an interconnection pattern using two-qubit gates (e.g., CNOT, SWAP, CZ etc.). For example, Fig 1. shows a chain pattern with two possible implementations of 2-qubit gate disjoint layers (see. Fig.2 and Fig 3.). Fig 2. implements the chain pattern in two distinct layers of depth-1 gates whereas Fig 3. implements it in three distinct layers.

Layer composition of CNOTs for a linear chain of qubits

An implementation is chosen (we choose the implementation of Fig. 2). A Direct Simultaneous Randomized Benchmarking is then run for each layer (see Fig. 4). The RB circuit is composed of \(l\) blocks with an ending gate \(g_{end}\) inverting the previous sequence of quantum gates. Each block is composed of a wall of random Clifford single-quit gate and of the depth-1 layer of CNOT gate, separated by barrieres.

Simultaneous direct randomized benchmarking protocol for each layer of CNOT

The RB protocol permits to extract the exponential single or joint decay factor \(\alpha_i\). A fidelity \(F_i\) is then computed from each decay factor with the formula. The disjoint layer fidelity \(LF_m\) is a product of each fidelity \(F_i\) and the final layer fidelity \(LF\) is the product of each disjoint layer fidelity. The EPLG is then computed from the final layer fidelity \(LF\) and the number of 2-qubit gate in all the layers \(n_{2q}\) (3 in our example).

As the number of qubits grows, the number of possible implementations for a selected pattern grows exponentially. The authors of [1] employ a heuristic to select the best implementation that produces the highest EPLG score (only the best EPLG score is reported).

Assumptions

The equation that computes \(LF_m\) is only valid if there is no cross-talk errors. The second expression defining \(LF\) defines a lower bound on the error and is only a good approximation if the error is low.
This protocol uses the same assumptions as standard RB protocols (for instance, that the noise is markovian).

Limitations

This protocol relies on simultaneous RB and hence, possess the same limitations.
The performance of the final computation strongly depends on the heuristic used to find good patterns that produce high EPLG.
This estimation is just a lower bound on the error rate and has to be coutiously interpreted.
The EPLG strongly depends on the heuristic used to select the best implementation of the pattern.

References

[1]D. C. McKay, I. Hincks, E. J. Pritchett, M. Carroll, L. C. G. Govia, and S. T. Merkel, “Benchmarking quantum processor performance at scale,” arXiv preprint arXiv:2311.05933, 2023.