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arXiv:2510.08313

Characterizing Space-Constrained Implementability of Quantum Instruments via Signaling Conditions

Matsui, Kosuke, Jun-Yi Wu, Hayata Yamasaki, Min-Hsiu Hsieh, Mio Murao

Published Oct 9, 2025• 0 citations
arXivHTMLPDF

Abstract

Scaling up the number of qubits available on quantum processors remains technically demanding even in the long term; it is therefore crucial to clarify the number of qubits required to implement a given quantum operation. For the most general class of quantum operations, known as quantum instruments, the qubit requirements are not well understood, especially when mid-circuit measurements and delayed input preparation are permitted. In this work, we characterize lower and upper bounds on the number of qubits required to implement a given quantum instrument in terms of the causal structure of the instrument. We further apply our results to entanglement distillation protocols based on stabilizer codes and show that, in these cases, the lower and upper bounds coincide, so the optimal qubit requirement is determined. In particular, we compute that the optimal number of qubits is 3 for the $[[9,1,3]]$-code-based protocol and 4 for the $[[5,1,3]]$-code-based protocol.

DOI
10.48550/arxiv.2510.08313

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Scaling up the number of qubits available on quantum processors remains technically demanding even in the long term; it is therefore crucial to clarify the number of qubits required to implement a given quantum operation. For the most general class of quantum operations, known as quantum instruments, the qubit requirements are not well understood, especially when mid-circuit measurements and delayed input preparation are permitted. In this work, we characterize lower and upper bounds on the number of qubits required to implement a given quantum instrument in terms of the causal structure of the instrument. We further apply our results to entanglement distillation protocols based on stabilizer codes and show that, in these cases, the lower and upper bounds coincide, so the optimal qubit requirement is determined. In particular, we compute that the optimal number of qubits is 3 for the [[9,1,3]][[9,1,3]][[9,1,3]]-code-based protocol and 4 for the [[5,1,3]][[5,1,3]][[5,1,3]]-code-based protocol.
In adaptive quantum circuits classical results of mid-circuit measurements determine the upcoming gates. This allows POVMs, quantum channels or more generally quantum instruments to be implemented sequentially, so that fewer qubits need to be used at each of the NNN measurement steps. In this paper, we mathematically describe these problems via adaptive sequence of instruments (ASI) and show how any instrument can be decomposed into it. Number of steps NNN and number of ancillary qubits nAn_AnA​ needed for actual implementation are crucial parameters of any such ASI. We show an achievable lower bound on the product N.nAN.n_AN.nA​ and we determine in which situations this tradeoff is likely to be optimal. Contrary to common intuition we show that for quantum instruments which transform nnn to m(>n)m(>n)m(>n) qubits, there exist NNN-step ASI implementing them just with (m−n)(m-n)(m−n) ancillary qubits, which are remeasured (N−1)(N-1)(N−1) times and finally used as output qubits.
We study space-bounded communication complexity for unitary implementation in distributed quantum processors, where we restrict the number of qubits per processor to ensure practical relevance and technical non-triviality. We model distributed quantum processors using distributed quantum circuits with nonlocal two-qubit gates, defining the distributed communication complexity of a unitary as the minimum number of such nonlocal gates required for its realization, up to permutations of data qubit positions. Our contributions are twofold. First, for general nnn-qubit unitaries, we improve upon the trivial O(4n)O(4^n)O(4n) communication bound. Considering kkk pairwise-connected processors (each with n/kn/kn/k data qubits and mmm ancillas), we prove the communication complexity satisfies O(max⁡{4(1−1/k)n−m,n})O\left(\max\{4^{(1-1/k)n - m}, n\}\right)O(max{4(1−1/k)n−m,n}) -- for example, O(2n)O(2^n)O(2n) when m=0m=0m=0 and k=2k=2k=2 -- and establish the tightness of this upper bound. We further extend the analysis to approximation models and general network topologies. Second, for special unitaries, we show that both the Quantum Fourier Transform (QFT) and Clifford circuits admit linear upper bounds on communication complexity in the exact model, outperforming the trivial quadratic bounds applicable to these cases. In the approximation model, QFT's communication complexity reduces drastically from linear to logarithmic, while Clifford circuits retain a linear lower bound. These results offer fundamental insights for optimizing communication in distributed quantum unitary implementation, advancing the feasibility of large-scale DQC systems.
Qubits are the minimum unit of information in quantum computing. As current quantum devices have a very limited number of available qubits, finding circuits that optimize their use is crucial for this computing paradigm. Optimizing the qubits needed to implement a quantum circuit is not trivial, as these circuits have important restrictions such as the impossibility of copying values or the requirement that the computation done in them must always be reversible. This paper presents a framework for optimizing circuits for quantum computing. The framework is able to obtain, given a quantum circuit, an equivalent one that is more efficient in terms of number of qubits. In particular, the framework has applicability in quantum circuits dedicated to arithmetic operations such as addition or multiplication. As a representative case study, it is applied to a reversible comparator circuit, obtaining a design that reduces the number of qubits and controlled gates compared to other reversible comparators reported in the recent literature. As part of the work, a public repository is included with the necessary code to simplify any other circuit.
We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of controlled-not (cnot) gates required (single-qubit gates are free). We consider three different models. In the first, the quantum circuit model (QCM), we consider sequences of single-qubit and cnot gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of cnot gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from mmm qubits to nnn qubits that uses at most one ancilla and has a low cnot count. We give explicit examples for small numbers of qubits that provide the lowest known cnot counts.
Statistical verification of a quantum state aims to certify whether a given unknown state is close to the target state with confidence. So far, sample-optimal verification protocols based on local measurements have been found only for disparate groups of states: bipartite pure states, GHZ states, and antisymmetric basis states. In this work, we investigate systematically optimal verification of entangled stabilizer states using Pauli measurements. First, we provide a lower bound on the sample complexity of any verification protocol based on separable measurements, which is independent of the number of qubits and the specific stabilizer state. Then we propose a simple algorithm for constructing optimal protocols based on Pauli measurements. Our calculations suggest that optimal protocols based on Pauli measurements can saturate the above bound for all entangled stabilizer states, and this claim is verified explicitly for states up to seven qubits. Similar results are derived when each party can choose only two measurement settings, say X and Z. Furthermore, by virtue of the chromatic number, we provide an upper bound for the minimum number of settings required to verify any graph state, which is expected to be tight. For experimentalists, optimal protocols and protocols with the minimum number of settings are explicitly provided for all equivalent classes of stabilizer states up to seven qubits. For theorists, general results on stabilizer states (including graph states in particular) and related structures derived here may be of independent interest beyond quantum state verification.
Recent constructions of quantum low-density parity-check (QLDPC) codes provide optimal scaling of the number of logical qubits and the minimum distance in terms of the code length, thereby opening the door to fault-tolerant quantum systems with minimal resource overhead. However, the hardware path from nearest-neighbor-connection-based topological codes to long-range-interaction-demanding QLDPC codes is likely a challenging one. Given the practical difficulty in building a monolithic architecture for quantum systems, such as computers, based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected medium-sized quantum processors. In such a setting, all syndrome measurements and logical operations must be performed through the use of high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. This protocol is applicable beyond the application of DQC since entanglement distribution and purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder with a sequential schedule for distilling 3 -qubit GHZ states using a rate 0.118 family of lifted product QLDPC codes and obtain an input fidelity threshold of ≈0.7974 under i.i.d. single-qubit depolarizing noise. This represents the best threshold for a yield of 0.118 for any GHZ purification protocol. Our results apply to larger size GHZ states as well, where we extend our technical result about a measurement property of 3 -qubit GHZ states to construct a scalable GHZ purification protocol.