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arXiv:2411.10024

NNN-Party Hadamard Test for Distributed Quantum Computation

Kaoru Yamamoto, Yuichiro Matsuzaki, Yasunari Suzuki, Yuuki Tokunaga, Suguru Endo

Published Nov 15, 2024• 0 citations
arXivHTMLPDF

Abstract

Quantum computers promise computational advantages over classical computers, but hardware-imposed limitations remain a major obstacle. The Hadamard test mitigates these limitations by estimating expectation values associated with resource-intensive quantum operations using simple quantum circuits at the cost of additional classical sampling, and therefore underlies many quantum algorithms. However, in distributed quantum computing (DQC), which offers a promising route to scalability, its use is hindered by the need for nonlocal controlled operations. Here we introduce an $N$-party Hadamard test for DQC that estimates the same expectation values as the standard Hadamard test without implementing nonlocal controlled operations. The protocol instead uses pre-shared entanglement together with local operations and classical communication, which are standard resources in DQC settings. To demonstrate its utility, we apply it to unitary operations for clustered Hamiltonian simulation and to projectors for stabilizer-state preparation, showing lower sampling overheads than previous approaches by exploiting pre-shared entangled ancilla states. Moreover, we numerically demonstrate Bell-state preparation from Werner states to show favorable sampling efficiency and noise robustness relative to conventional purification, circuit knitting/cutting, and probabilistic error cancellation. Our work provides a general strategy for bringing Hadamard-test-based algorithms to DQC, facilitating practical and flexible quantum computation.

DOI
10.48550/arxiv.2411.10024

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Quantum computers promise computational advantages over classical computers, but hardware-imposed limitations remain a major obstacle. The Hadamard test mitigates these limitations by estimating expectation values associated with resource-intensive quantum operations using simple quantum circuits at the cost of additional classical sampling, and therefore underlies many quantum algorithms. However, in distributed quantum computing (DQC), which offers a promising route to scalability, its use is hindered by the need for nonlocal controlled operations. Here we introduce an NNN-party Hadamard test for DQC that estimates the same expectation values as the standard Hadamard test without implementing nonlocal controlled operations. The protocol instead uses pre-shared entanglement together with local operations and classical communication, which are standard resources in DQC settings. To demonstrate its utility, we apply it to unitary operations for clustered Hamiltonian simulation and to projectors for stabilizer-state preparation, showing lower sampling overheads than previous approaches by exploiting pre-shared entangled ancilla states. Moreover, we numerically demonstrate Bell-state preparation from Werner states to show favorable sampling efficiency and noise robustness relative to conventional purification, circuit knitting/cutting, and probabilistic error cancellation. Our work provides a general strategy for bringing Hadamard-test-based algorithms to DQC, facilitating practical and flexible quantum computation.
In noisy intermediate-scale quantum computing, the limited scalability of a single quantum processing unit (QPU) can be extended through distributed quantum computing (DQC), in which one can implement global operations over two QPUs by entanglement-assisted local operations and classical communication. To facilitate this type of DQC in experiments, we need an entanglement-efficient protocol. To this end, we extend the protocol in [Eisert et. al., PRA, 62:052317(2000)] implementing each nonlocal controlled-unitary gate locally with one maximally entangled pair to a packing protocol, which can pack multiple nonlocal controlled-unitary gates locally using one maximally entangled pair. In particular, two types of packing processes are introduced as the building blocks, namely the distributing processes and embedding processes. Each distributing process distributes corresponding gates locally with one entangled pair. The efficiency of entanglement is then enhanced by embedding processes, which merge two non-sequential distributing processes and hence save the entanglement cost. We show that the structure of distributability and embeddability of a quantum circuit can be fully represented by the corresponding packing graphs and conflict graphs. Based on these graphs, we derive heuristic algorithms for finding an entanglement-efficient packing of distributing processes for a given quantum circuit to be implemented by two parties. These algorithms can determine the required number of local auxiliary qubits in the DQC. We apply these algorithms for bipartite DQC of unitary coupled-cluster circuits and find a significant reduction of entanglement cost through embeddings. This method can determine a constructive upper bound on the entanglement cost for the DQC of quantum circuits.
Distributed quantum computing relies on coordinated operations between remote quantum processing units (QPUs), yet most existing work either assumes full connectivity, unrealistic for large networks, or relies on entanglement swapping. To mitigate the overhead of communication, we propose a scheme for the distribution of collective quantum operations among remote quantum processing units by exploiting distributed fan-out operations to a central node in network architectures similar to those used for high-performance computing, which requires only pre-shared entanglement, local operations and classical communication. We show that a general diagonal gate can be distributed among any number of nodes and provide the ebit cost bounds. For a single distributed multicontrolled gate, this amounts to a single additional Bell pair over the theoretically optimal calculation with all-to-all pre-shared entanglement, demonstrating better scalability when compared to current proposals based on entanglement swapping through a network. We provide a recipe for the lumped distribution of gates such as arbitrarily-sized Toffoli and multicontrolled Z, andRzz(θ)R_{zz}(\theta)Rzz​(θ)gates. Finally, we provide an exact implementation of a distributed Grover's search algorithm using this protocol to partition the circuit, with Bell pair cost growing linearly with the number of Grover iterations and the number of partitions, and show how these techniques can be applied to other algorithms such as QAOA. Our results show that alternative approaches to entanglement swapping can provide major benefits in distributed quantum computing, pointing to promising avenues for future research.
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The Hadamard test is a fundamental building block widely used in many quantum computing algorithms. It estimates the real or imaginary part of ⟨ψ∣U∣ψ⟩\langle ψ\vert U \vert ψ\rangle⟨ψ∣U∣ψ⟩, where ∣ψ⟩\vert ψ\rangle∣ψ⟩ is a quantum state and UUU is a unitary operator. In many algorithms, however, many such quantities must be estimated, leading to a large number of distinct circuit types, long computational times, and high financial costs. In this work, we propose the parallel Hadamard test, which combines multiple Hadamard tests into a single circuit type. We demonstrate how the parallel Hadamard test applies to three structural classes of workloads: arbitrary sets of unitary operators, prefix-product arrays, and Gram-matrix elements. For each class, we compare the cost of the parallel Hadamard test with that of the conventional one. Our unified approach significantly reduces the number of distinct circuit types, and can lower both computational time and financial costs in regimes where fixed per-circuit overheads dominate the total cost. In Gram-matrix workloads, it can also reduce the total number of shots when typical off-diagonal overlaps are small.
With the rapid advancement of quantum information technology, designing efficient distributed quantum algorithms to perform various information processing tasks remains challenging. In this paper, we consider a distributed scenario where two parties, Alice and Bob, given access to matrices AAA and BBB respectively, aim to estimate Tr(f(A)g(B)){\rm Tr}(f(A)g(B))Tr(f(A)g(B)), where fff and ggg are known functions. In this task, only local quantum operations, classical communication and single-qubit measurements are allowed due to the high cost of quantum communication and entangled measurements. We propose a distributed quantum algorithm framework to compute Tr(f(A)g(B)){\rm Tr}(f(A)g(B))Tr(f(A)g(B)) within an additive error ε\varepsilonε. The proposed algorithm framework requires O~(d2/(δε2))\widetilde{O}\left(d^2 / (δ\varepsilon^2)\right)O(d2/(δε2)) quantum queries and two-qubit gates, assuming that the minimum singular value of each matrix A,B∈Cd×dA, B \in \mathbb{C}^{d \times d}A,B∈Cd×d is at least δδδ. Additionally, our algorithm framework uses a simple Hadamard test architecture, enabling easier quantum hardware implementation. This algorithm framework allows for various applications including quantum divergence estimation, distributed solving of linear system, and distributed Hamiltonian simulation, while preserving the privacy of both parties. Furthermore, we establish a lower bound of Ω(max⁡{1/ε2,dr/ε})Ω\left(\max\left\{1 /\varepsilon^2, \sqrt{dr}/\varepsilon\right\}\right)Ω(max{1/ε2,dr​/ε}) on the query complexity for the task, where rrr denotes the rank of the input matrices. We believe this framework holds broad applicability across a range of distributed quantum computing tasks.
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