Quantum State Estimation Achieves Efficiency with Low-Depth Circuit Designs.

The efficient determination of the inner product between quantum states distributed across separate locations represents a core challenge in distributed quantum information processing, with implications for secure communication and networked quantum computation. Researchers are actively seeking methods to minimise the resources required for this task, particularly given the limitations of current quantum hardware. A team comprising Congcong Zheng, Kun Wang, Xutao Yu, Ping Xu, and Zaichen Zhang, affiliated with Southeast University and the National University of Defence Technology, now present a detailed analysis of distributed inner product estimation utilising quantum circuits of limited depth. Their work, entitled “Distributed Quantum Inner Product Estimation with Low-Depth Circuits”, establishes theoretical bounds on the number of quantum measurements needed, demonstrating that performance comparable to designs requiring more complex circuits can be achieved with experimentally viable alternatives such as brickwork and Clifford ensembles. The analysis considers both average-case and state-dependent performance, offering insights into the practical implementation of this fundamental quantum task.

Distributed inner product estimation (DIPE) represents a fundamental task within quantum information science, enabling the determination of the inner product between quantum states held on separate quantum platforms. Rigorous analyses of DIPE’s sample complexity, which dictates the number of measurements required for accurate estimation, traditionally rely on the use of unitary t-designs. A unitary t-design is an ensemble of unitary circuits that mimics the behaviour of a completely random unitary transformation up to order t, but their implementation poses substantial challenges for near-term quantum devices due to their inherent circuit depth. Recent research addresses this limitation by investigating DIPE utilising circuits of limited depth, offering a potential route towards experimentally feasible implementations.

The work establishes that employing an arbitrary unitary t-design ensemble achieves an average sample complexity of O(t), where t denotes the number of qubits. This provides a benchmark against which to compare more practical, low-depth circuit ensembles, allowing researchers to quantify the trade-offs involved in simplifying circuit designs. The analysis extends to ensembles that fall short of the stringent requirements of full unitary t-designs, demonstrating a commitment to finding solutions compatible with current technological limitations. Specifically, brickwork and local unitary t-design ensembles exhibit average sample complexities of O(t²) and O(t³), respectively, indicating a reduction in computational cost.

Further investigation focuses on the state-dependent variance observed when utilising the brickwork and Clifford ensembles. The research reveals a notable finding: DIPE employing the global Clifford ensemble requires only O(t) copies for arbitrary state pairs, with its efficiency further enhanced when the state pairs exhibit non-stabilizerness. Stabilizerness describes the extent to which a group of stabiliser operators can describe a quantum state; non-stabilizer states generally require more resources for processing, making this finding particularly significant. This suggests that careful selection of quantum states can further optimise DIPE protocols.

Numerical simulations, conducted with Greenberger–Horne–Zeilinger (GHZ) states and Haar random states up to 12 qubits, demonstrate that these low-depth circuit ensembles can, in practice, match the performance of full unitary t-designs. GHZ states are a specific type of entangled quantum state, while Haar random states represent a uniformly distributed ensemble of quantum states. These simulations suggest that these ensembles offer a viable pathway for implementing DIPE on current and near-future quantum hardware, and validate the theoretical predictions with experimental results.

This research establishes a clear path towards more efficient and practical DIPE protocols, paving the way for advancements in quantum communication and computation. Researchers are actively exploring further optimisations and extensions of these techniques, aiming to unlock the full potential of quantum information processing. The findings presented here represent a significant step forward in bridging the gap between theoretical concepts and real-world implementations of quantum technologies.