Extended Quantum Computing Patterns

How to partition a multi-qubit gate into independent gates while preserving the computation's result?

A quantum circuit contains a multi-qubit gate operating between two partitions of qubits and cannot be expressed as a single product of local gates, i.e., gates that operate exclusively on qubits within a single partition. This gate can create entanglement between the partitions, and its execution requires direct physical connections between the involved qubits. The focus is on the expectation value across multiple shots, not single-shot experiments.

Limited connectivity between qubits of the multi-qubit gate requires compensating with SWAP operations that exchange the states of two qubits to establish required connections, thereby introducing additional errors [Leymann et al., 2020]. The complete absence of connectivity between the qubit partitions, e.g., multiple devices without quantum communication, renders the execution of gates between partitions impossible.

Apply a gate cut to decompose a circuit's multi-qubit gate into a weighted sum of subcircuits Mitarai et al., 2021. As shown in the solution sketch, in each subcircuit, the original multi-qubit gate \(U\) is replaced by local operations \(F_{i}^{A}\) and \(F_{i}^{B}\), which act on two disjoint qubit partitions labeled \(A\) and \(B\). The sum of the expectation values of these subcircuits, each weighted by real coefficient \(a_i\), must replicate the expectation value of the original circuit with gate \(U\). The local operations \(F_{i}^{A}\) and \(F_{i}^{B}\) can be either unitary transformations or projective measurements. Execute these subcircuits and combine their results using the real coefficients \(a_{i}\).

The GATE CUT enables implementing a multi-qubit gate as a linear combination of local operations independent of the qubit connectivity. This eliminates the need for additional SWAP gates when applied on a single device and allows a multi-qubit gate across two quantum devices without requiring quantum communication. However, replacing multi-qubit gates with local gates and projections through executing multiple subcircuits increases the computational overhead in terms of required shots to achieve the same statistical accuracy in the expectation value as the original circuit. To reduce computational overhead when cutting multiple gates, consider incorporating classical communication between the partitions [Piveteau et al, 2024]. Executing subcircuits in parallel on multiple devices reduces the overall computation runtime [Bravyi et al, 2022]. Moreover, this approach reproduces only the expectation value, making it unsuitable for single-shot experiments [Piveteau et al, 2024].

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The GATE CUT pattern is used in CIRCUIT CUTTING. PRIORITIZED EXECUTION can speed up subcircuit execution. The POST-SELECTIVE MEASUREMENT pattern can handle local projective measurements. Additionally, a GATE CUT can be employed in VQAs, such as VQE or QAOA, to reduce ansatz connectivity without creating disconnected subcircuits.

Gate cuts are used in various circuit cutting works, including two-qubit gate cutting [Mitarai et al., 2021, Bechtold et al.,2023] and many-qubit gate cutting [Ufrecht et al., 2023]. Moreover, gate cuts are applied to reduce errors by minimizing the number of two-qubit gates [Yamamoto et al., 2023]. An implementation is provided in Qiskit's Circuit Knitting Toolbox.