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Satoshi Yoshida

Publications and source records attributed to Satoshi Yoshida.

At least 19 recordsLinked to original sources
arXiv2026

Barycentric programming: Classical realization of programs for universal quantum processors

A programmable quantum processor implements a target unitary using information stored in a classical or quantum program register. For arbitrary unitaries in $\mathrm{SU}(d)$, quantum programs attain the optimal leading-order memory cost, whereas deterministic classical rounding requires twice as much memory for the same diamond-distance error. We show that this gap disappears when classical randomization is allowed. We introduce barycentric programming, which stores the label of a unitary randomly selected from a finite set, with probabilities chosen so that first-order approximation errors cancel in the averaged unitary channel. Consequently, the diamond-distance error scales quadratically with the discretization spacing. For any $ν$-parameter family of unitaries forming a compact, smooth manifold without boundary, this method achieves error $\varepsilon$ using only $(ν/2)\log_2(1/\varepsilon)+O(1)$ classical bits. We also prove a matching lower bound for arbitrary quantum programs, determining the optimal leading-order program size for general unitary families. Thus, classical randomization achieves the optimal scaling $(ν/2+o(1))\log_2(1/\varepsilon)$ without requiring coherent quantum memory.

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arXiv2026

Advantage of Sample Complexity in Quantum PAC Learning Requires Inverse Access to State-Preparation Unitaries

Whether quantum computation can reduce the amount of data required for conventional machine learning is a central problem in quantum machine learning. In probably approximately correct (PAC) learning, a learner receives training examples sampled from an unknown probability distribution. Quantum PAC learning extends this framework by allowing access to quantum examples, which are quantum states that coherently encode this distribution, or to unitaries that prepare quantum examples, with or without inverse access. When only copies of quantum examples are available, the optimal worst-case sample complexity asymptotically matches that of classical learning. In contrast, access to both a preparation unitary and its inverse can improve the accuracy dependence of query complexity. However, it has remained open whether access to the preparation unitary alone suffices for this improvement. In this work, we resolve this question by determining the optimal query complexity with forward-only access, taking the worst case over compatible preparation unitaries and their dimensions. In both realizable and agnostic PAC learning, this complexity matches, up to constant factors, the optimal sample complexity achievable with classical examples or copies of quantum examples. To prove these results, we construct, for any algorithm with forward-only access, a simulator that approximates the algorithm's Haar-averaged output state using as many copies of the prepared state as the algorithm makes queries. These results show that inverse access is essential for the known improvement in accuracy dependence in this worst-case setting. Beyond PAC learning, our reduction provides a general tool for transferring lower bounds for state copies to query lower bounds for forward-only access to the corresponding preparation unitaries.

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arXiv2026

Optimal learning of covariant quantum states and channels

We give collective tomography protocols for quantum states and channels with known symmetries, using random purification and dilation to reduce learning to pure-state estimation. For states commuting with a compact-group representation with multiplicities $m_λ$, the optimal copy complexity is $Θ((\sum_λm_λ^2+\logη^{-1})/\varepsilon^2)$ for sufficiently small trace-distance error $\varepsilon$ and failure probability $η$ for the nontrivial case $\sum_λm_λ^2>1$. For $G$-covariant channels with finite-dimensional unitary representations of a compact group $G$, parallel $\widetilde{O}(D_G/\varepsilon^2)$ queries achieve a diamond-distance error $\varepsilon $ at fixed success probability $2/3$, where $D_G$ counts the real parameters of covariant Choi operators before imposing trace preservation. For permutation-covariant channels on $k$ qudits of fixed local dimension $d$, at sufficiently small fixed error and fixed success probability $2/3$, we obtain the optimal query scalings $Θ_d(k^{d^4-1})$ for diamond distance and $Θ_d(k^{d^4-d^2})$ for Choi trace distance. Furthermore, we resolve an open problem in quantum tomography of constructing efficient quantum circuits that approximate the Hayashi measurement for optimal pure-state estimation. We encode states in the symmetric subspace into bosonic occupation modes, realize the measurement by heterodyne detection and normalization, and approximate this procedure on qubits using the quantum Hermite transform. Combined with our symmetry-compatible purification and dilation circuits, this yields query-optimal and gate-efficient learners of permutation-covariant states and channels with gate complexity $O_d(\mathrm{poly}(k,\varepsilon^{-1},\logη^{-1}))$.

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arXiv2026

An exponential query advantage from access to Stinespring dilation unitaries in quantum channel learning

We present a learning task for quantum channels that exhibits an exponential separation in query complexity (in terms of the number of qubits) between two access models: i) black-box channel access and ii) access to a Stinespring dilation unitary of the channel along with its inverse. We consider the task of learning how close the unknown channel is to the set of unitary channels. This is characterized by the \textit{optimal average gate fidelity (OAGF)}, defined as the average gate fidelity maximized over all unitary channels. For constant Kraus rank and a dilation environment of dimension $\operatorname{polylog}(d)$, we provide an algorithm that estimates the OAGF to fixed additive accuracy using $\operatorname{polylog}(d)$ queries to the Stinespring dilation unitary and its inverse. In contradistinction, under the black-box access model, this estimation requires at least $Ω(\sqrt{d})$ queries, even for channels of constant Kraus rank and constant additive accuracy. As a corollary, we rule out a universal simulator that uses only $\operatorname{poly}(\log d,q)$ queries to the unknown channel to approximate the output of arbitrary $q$-query algorithms with access to a randomly-sampled Stinespring dilation unitary and its inverse, even to sufficiently small constant error. These results demonstrate that access to the system-environment joint evolution and its inverse can provide an exponential advantage in estimating an intrinsic property of a quantum channel.

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arXiv2026

Asymptotically optimal purification of noisy unitary channels in any dimension

We consider the problem of noisy unitary purification. Given access to an unknown $d$-dimensional unitary channel followed by depolarizing noise of strength $p$, we aim to construct a superchannel that universally purifies the noisy unitary back to the original unknown unitary. We optimize over arbitrary adaptive sequential strategies and analytically derive the optimal fidelity to the leading order in the noise strength and number of channel uses, while also providing a concrete $\mathrm{SU}(d)$-covariant parallel strategy that attains the optimum. Our result implies the query complexity $Θ(d^2p/ε)$ for achieving leading-order infidelity $ε$ in the low-noise regime, which scales better than the naive approach combining optimal state purification and storage-and-retrieval of quantum channels. We also consider the dual problem of noisy unitary conjugation, where the goal is to obtain the best approximation of the complex conjugate of the original unknown unitary from access to noisy queries. We show that the optimal fidelity for this task coincides with that of noisy unitary purification to the leading-order in the low-noise and large-query limit.

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arXiv2026

Optimal complex conjugation of unknown isometry channels

Access to the complex conjugate of an unknown quantum channel is a useful resource in quantum oracle problems, motivating the question of how such access can be simulated using only a limited number of calls to the original channel. We determine the optimal deterministic protocol for approximately implementing the complex conjugate isometry $\overline{V}$ from $n$ uses of an unknown isometry channel $V: \mathbb{C}^d\to\mathbb{C}^D$. We derive a closed-form expression for the optimal fidelity and prove that a parallel protocol is optimal even among general quantum superchannels, including adaptive and indefinite-causal-order strategies. The formula implies a query complexity $n=Θ(d[(D-d)/ε+1])$ for achieving infidelity $ε$. We also present a circuit construction based on the quantum Schur transform and the dual Clebsch--Gordan transform, with circuit complexity $O(\mathrm{poly}(D,1/ε))$. This task is extended to the multi-copy case $V^{\otimes n}\mapsto \overline{V}^{\otimes k}$. For fixed $d<D$ and $k$, we show that the optimal fidelity for the multi-copy case is $1-kd(D-d)/n+o(n^{-1})$, and that this value is asymptotically attained by a parallel estimation-based protocol. Finally, combining the isometry protocol with random Stinespring dilations yields a protocol for complex conjugation of unknown rank-$r$ quantum channels whose query complexity is optimal up to a constant factor if the Kraus rank $r$ is constant.

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arXiv2026

Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms

We construct exact compressed oracles for Haar-random group actions associated with an arbitrary finite-dimensional unitary representation of a compact group. The construction is a representation-theoretic version of Zhandry's compressed-oracle technique: the memory of the oracle is stored in the Fourier basis, and each update is implemented by Clebsch-Gordan transforms. This framework naturally gives forward, conjugate, transpose, and inverse compressed oracles. For the unitary group with the defining representation, we present efficient implementation based on high-dimensional Clebsch-Gordan transforms. We also explain how Ma-Huang's approximate path-recording oracle compares to our exact construction. For general compact groups, we describe the corresponding path-recording bases, achieved via generalized Schur transforms. These results clarify the relation between exact representation-theoretic compressed oracles and the path-recording bases used in algorithmic and cryptographic compressed-oracle arguments.

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arXiv2026

Scaling-optimal purification of noisy qubit unitary channels

We consider the problem of purifying noisy qubit unitary channels. Given the ability to apply an unknown qubit unitary channel followed by depolarizing noise, we aim to construct a superchannel that purifies the noisy unitary back to the original unknown unitary. We first provide numerical evidence that sequential strategies can strictly outperform parallel strategies when the number of channel uses is finite, highlighting the fundamental distinction from state purification. We then provide a concrete $\mathrm{U}(2)$-covariant parallel protocol based on a novel entanglement-assisted quantum error-correcting code that suppresses the first-order noise strength as $O(1/n)$ with $n$ channel uses and show this scaling is asymptotically optimal in the low-noise regime, even when sequential strategies are allowed.

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arXiv2026

Sequential quantum processes with group symmetries

Symmetry plays a crucial role in the design and analysis of quantum protocols. This result shows a canonical circuit decomposition of a $(G\times H)$-invariant quantum comb for compact groups $G$ and $H$ using the corresponding Clebsch--Gordan transforms, which naturally extends to the $G$-covariant quantum comb. By using this circuit decomposition, we propose a parametrized quantum comb with group symmetry, and derive the optimal quantum comb which transforms an unknown unitary operation $U\in \mathrm{SU}(d)$ into its inverse $U^\dagger$ or transpose $U^\top$. From numerics, we find a deterministic and exact unitary transposition protocol for $d=3$ with $7$ queries to $U$. This protocol improves upon the protocol shown in the previous work, which requires $13$ queries to $U$.

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arXiv2026

Random dilation superchannel

We present a quantum circuit that implements the random dilation superchannel, transforming parallel queries of an unknown quantum channel into the same number of parallel queries of a randomly chosen dilation isometry of the input channel. This is a natural generalization of the random purification channel, that transforms copies of an unknown mixed state to copies of a randomly chosen purification state. The circuit complexity of our construction is $O(\mathrm{poly}(n, \log d_I, \log d_O))$, where $n$ is the number of queries and $d_I$ and $d_O$ are the input and output dimensions of the input channel, respectively. This random dilation superchannel is extended to the sequential queries approximately, by transforming the parallel random dilation isometry into sequential random dilation unitaries with $O(\mathrm{poly}(d_I))$ overhead in the number of queries. We also show that our results can be further extended to the case of quantum superchannels. On the other hand, we show a no-go theorem on the exact random dilation of sequential queries with $o(\mathrm{poly}(\min\{d_I, d_O\}))$ query overhead, showcasing a fundamental difference between the parallel and sequential cases. As an application, we show an efficient storage-and-retrieval of an unknown quantum channel, which improves the program cost exponentially in the retrieval error $\varepsilon$. For the case where the Kraus rank $r$ is the least possible (i.e., $r = d_I/d_O$), we show quantum circuits that transform $n$ parallel queries of an unknown quantum channel $Λ$ to $Θ(n^α)$ parallel queries of $Λ$ for any $α<2$ approximately, and implement its Petz recovery map for the maximally mixed reference state probabilistically and exactly.

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arXiv2026

Proof of a finite threshold for the union-find decoder

Fast decoders that achieve strong error suppression are essential for fault-tolerant quantum computation (FTQC) from both practical and theoretical perspectives. The union-find (UF) decoder for the surface code is widely regarded as a promising candidate, offering almost-linear time complexity and favorable empirical error suppression supported by numerical evidence. However, the lack of a rigorous threshold theorem has left open whether the UF decoder can achieve fault tolerance beyond the error models and parameter regimes tested in numerical simulations. Here, we provide a rigorous proof of a finite threshold for the UF decoder on the surface code under the circuit-level local stochastic error model. To this end, we develop a refined error-clustering framework that extends techniques previously used to analyze cellular-automaton and renormalization-group decoders, by showing that error clusters can be separated by substantially larger buffers, thereby enabling analytical control over the behavior of the UF decoder. Using this guarantee, we further prove a quasi-polylogarithmic upper bound on the average runtime of a parallel UF decoder in terms of the code size. We also show that this framework yields a finite threshold for the greedy decoder, a simpler decoder with lower complexity but weaker empirical error suppression. These results provide a solid theoretical foundation for the practical use of UF-based decoders in the development of fault-tolerant quantum computers, while offering a unified framework for studying fault tolerance across these practical decoders.

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arXiv2026

Multicopy quantum state teleportation with application to storage and retrieval of quantum programs

This work considers a teleportation task for Alice and Bob in a scenario where Bob cannot perform corrections. In particular, we analyse the task of \textit{multicopy state teleportation}, where Alice has $k$ identical copies of an arbitrary unknown $d$-dimensional qudit state $\vertψ\rangle$ to teleport a single copy of $\vertψ\rangle$ to Bob using a maximally entangled two-qudit state shared between Alice and Bob without Bob's correction. Alice may perform a joint measurement on her half of the entangled state and the $k$ copies of $\vertψ\rangle$. We prove that the maximal probability of success for teleporting the exact state $\vertψ\rangle$ to Bob is $p(d,k)=\frac{k}{d(k-1+d)}$ and present an explicit protocol to attain this performance. Then, by utilising $k$ copies of an arbitrary target state $\vertψ\rangle$, we show how the multicopy state teleportation protocol can be employed to enhance the success probability of storage and retrieval of quantum programs, which aims to universally retrieve the action of an arbitrary quantum channel that is stored in a state. Our proofs make use of group representation theory methods, which may find applications beyond the problems addressed in this work.

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arXiv2026

Quantum Advantage in Storage and Retrieval of Isometry Channels

Storage and retrieval refer to the task of encoding an unknown quantum channel $Λ$ into a quantum state, known as the program state, such that the channel can later be retrieved. There are two strategies for this task: classical and quantum strategies. The classical strategy uses multiple queries to $Λ$ to estimate $Λ$ and retrieves the channel based on the estimate represented in classical bits. The classical strategy turns out to offer the optimal performance for the storage and retrieval of unitary channels. In this work, we analyze the asymptotic performance of the classical and quantum strategies for the storage and retrieval of isometry channels. We show that the optimal fidelity for isometry estimation is given by $F = 1-{d(D-d)\over n} + O(n^{-2})$, where $d$ and $D$ denote the input and output dimensions of the isometry, and $n$ is the number of queries. This result indicates that, unlike in the case of unitary channels, the classical strategy is suboptimal for the storage and retrieval of isometry channels, which requires $n = Θ(ε^{-1})$ to achieve the diamond-norm error $ε$. We propose a more efficient quantum strategy based on port-based teleportation, which stores the isometry channel in a program state using only $n = Θ(1/\sqrtε)$ queries, achieving a quadratic improvement over the classical strategy. As an application, we extend our approach to general quantum channels, achieving improved program cost compared to prior results by Gschwendtner, Bluhm, and Winter [Quantum \textbf{5}, 488 (2021)].

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arXiv2026

One-to-One Correspondence between Deterministic Port-Based Teleportation and Unitary Estimation

Port-based teleportation is a variant of quantum teleportation, where the receiver can choose one of the ports in his part of the entangled state shared with the sender, but cannot apply other recovery operations. We show that the optimal fidelity of deterministic port-based teleportation (dPBT) using $N=n+1$ ports to teleport a $d$-dimensional state is equivalent to the optimal fidelity of $d$-dimensional unitary estimation using $n$ calls of the input unitary operation. From any given dPBT, we can explicitly construct the corresponding unitary estimation protocol achieving the same optimal fidelity, and vice versa. Using the obtained one-to-one correspondence between dPBT and unitary estimation, we derive the asymptotic optimal fidelity of port-based teleportation given by $1-O(d^4)N^{-2}\leq F \leq 1-Ω(d^4)N^{-2}$, which improves the previously known result given by $1-O(d^5)N^{-2} \leq F \leq 1-Ω(d^2) N^{-2}$. We also show that the optimal fidelity of unitary estimation for the case $n\leq d-1$ is $F = {n+1 \over d^2}$, and this fidelity is equal to the optimal fidelity of unitary inversion with $n\leq d-1$ calls of the input unitary operation even if we allow indefinite causal order among the calls.

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arXiv2025

Simulating the quantum switch with quantum circuits is computationally hard

Higher-order transformations acting on input quantum channels in an indefinite causal order, such as the quantum switch, cannot be described by quantum circuits using the same number of calls to the input channels. A natural question is whether they can be simulated, i.e., whether their action can be exactly and deterministically reproduced by a quantum circuit with more calls to the input channels. Here, we prove that the quantum switch acting on two $n$-qubit channels cannot be simulated by any quantum circuit using $k$ calls to one channel and one to the other, if $k<2^n$. This establishes an exponential separation in quantum query complexity between processes with indefinite causal order and quantum circuits. Moreover, even with one extra call to both input channels, such a simulation remains impossible. We further demonstrate the robustness of this separation by extending the result to probabilistic and approximate simulations scenarios.

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arXiv2025

Analytical Lower Bound on Query Complexity for Transformations of Unknown Unitary Operations

Recent developments have revealed deterministic and exact protocols for performing complex conjugation, inversion, and transposition of a general $d$-dimensional unknown unitary operation using a finite number of queries to a black-box unitary operation. In this work, we establish analytical lower bounds for the query complexity of unitary inversion, transposition, and complex conjugation, which hold even if the input unitary is an unknown logarithmic-depth unitary. Specifically, our lower bound of $d^2$ for unitary inversion demonstrates the asymptotic optimality of the deterministic exact inversion protocol, which operates with $O(d^2)$ queries. We introduce a novel framework utilizing differentiation to derive these lower bounds on query complexity for general differentiable functions $f: \mathrm{SU}(d)\to \mathrm{SU}(d)$. As a corollary, we prove that a catalytic protocol -- a new concept recently noted in the study of exact unitary inversion -- is impossible for unitary complex conjugation. Furthermore, we extend our framework to the partially known setting, where the input unitary operation is promised to be within a subgroup of $\mathrm{SU}(d)$ and the probabilistic setting, where transformations succeed probabilistically.

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arXiv2025

Higher-order quantum computing with known input states

In higher-order quantum computing (HOQC), one typically considers the universal transformation of unknown quantum operations, treated as blackboxes. It is also implicitly assumed that the resulting operation must act on arbitrary, and thus unknown, input states. In this work, we explore a variant of this framework in which the operation remains unknown, but the input state is fixed and known. We argue that this assumption is well-motivated in certain practical contexts, such as unitary programming, and show that classical knowledge of the input state can significantly enhance performance. We demonstrate that in the SAR protocol, this knowledge leads to an exponential advantage through a repeat-until-success strategy, highlighting the operational power of known-state higher-order transformations. Moreover, this assumption allows us to distinguish between protocols designed for pure, bipartite, and mixed states, which enables us to identify the class of mixed states for which deterministic and exact implementation becomes possible.

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arXiv2025

Low Depth Color Code Circuits with CXSWAP gate

We present two new types of syndrome extraction circuits for the color code. Our first construction, which after [M. McEwen, D. Bacon, and C. Gidney, Quantum 7, 1172 (2023)] we call the semi-wiggling color code, promises to mitigate leakage errors by periodically interchanging the roles of bulk data and measurement qubits. The second construction reduces circuit depth relative to [C. Gidney and C. Jones, arXiv:2312.08813 (2023)] by employing the CXSWAP gate instead of CNOT. This optimization leads to $\sim 10\%$ improvement in teraquop footprint under the uniform error model with the physical error rate $p=0.1\%$.

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