Extended Quantum Computing Patterns

How to interrupt a wire in a quantum circuit classically such that no quantum information is transmitted while preserving the computation's result?

In a quantum circuit, a wire transfers a non-trivial quantum state between two gates, that is, a state that exhibits superposition and has the potential of entanglement with other qubits. The precise state conveyed by the wire is unknown. Furthermore, the quantum state experiences decay over time. Moreover, the interest of the experiment lies in the expectation value over multiple shots rather than the outcome of a single-shot execution.

Given the non-trivial nature of the state, it cannot be measured and reinitialized based on the measurement outcome, as the qubit's full state is not revealed through the measurement process. Instead, the measurement collapses the qubit probabilistically to a state associated with the measurement basis, destroying its superposition and entanglement. Furthermore, since the state of the wire is unknown, it cannot be restored independently of the previous measurement either.

Apply a wire cut to decompose a circuit's wire into a weighted sum of subcircuits [Peng et al., 2020]. As shown in the solution sketch, the subcircuits interrupt the wire with different measurements described by observables \(O_{i}\) and subsequent qubit initializations of the states \(\ket{\psi_{i}}\). 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. To cut the wire, run the subcircuits and combine their results based on the coefficients \(a_{i}\) and the measurement outcomes.

Applying a WIRE CUT interrupts the wire classically without disturbing the computation's expectation value. The behavior of superposition and entanglement is simulated by replacing the wire with a linear combination of subcircuits that perform a measurement and then reinitialize the qubit. However, the trade-off involves executing multiple subcircuits, which raises the computational overhead in terms of shots needed to maintain the same statistical accuracy in the computed expectation value as the original circuit. To reduce this computational overhead, it is advantageous to incorporate classical communication in the cut, enabling each state preparation to depend on its immediate prior measurement outcome [Brenner et al., 2023, Pednault, 2023, Harada et al., 2023]. This approach also lowers the cost of cutting multiple wires together compared to cutting each wire individually. Furthermore, leveraging multiple devices for parallel execution of subcircuits decreases the total computation runtime [Bravyi et al, 2022]. Additionally, employing maximum-likelihood fragment tomography can enhance subcircuit results and thereby improve the overall output quality [Perlin et al, 2021]. However, a wire cut only reproduces the result regarding the expectation value, and therefore, it is not suited for single-shot experiments [Brenner et al., 2023].

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The WIRE CUT pattern, used in CIRCUIT CUTTING, applies STATE PREPARATION for qubit initialization after measurement in the cut. Additionally, it can be employed in VQAs, e.g, VQE or QAOA. PRIORITIZED EXECUTION can speed up subcircuit execution.

Wire cutting is applied in several circuit cutting works [Tang et al., 2021, Lowe et al., 2023]. Moreover, their implementations are part of Pennylane and in Qiskit's Circuit Knitting Toolbox.