arXiv subjects

Dmitry Grinko

Publications and source records attributed to Dmitry Grinko.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Fourier transform for the symmetric group

Quantum Fourier transforms (QFT) for general groups were recognized to be fundamental already early in the field. A canonical example of non-abelian QFT for the symmetric group was outlined by Beals (1997). Later, a more detailed analysis of this algorithm was carried out by Kawano and Sekigawa (2016). In this paper, we revisit that construction. After a careful analysis, we revise their gate complexity to $\widetilde{\mathcal{O}}(n^{3.5})$ and circuit depth to $\widetilde{\mathcal{O}}(n^3)$. Moreover, we observe that their construction is not optimal in the choice of transversal elements, so we propose simpler realization of the symmetric group QFT.

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arXiv2026

Quantum Fourier transform toolbox

Quantum Fourier transforms (QFTs) are essential primitives in quantum algorithms. While abelian groups admit efficient QFT circuits, with circuit size polynomial in the logarithm of the group order, efficient constructions are known for relatively few non-abelian families. We develop two new approaches to QFT circuit construction, based on Mackey theory and Clifford theory, respectively, and use them to show exponential improvement in circuit cost for specific group families. Using the Mackey-theoretic approach, we obtain explicit quantum circuits for the QFT over $\mathrm{GL}_2(F_q)$ that scale polynomially in $\log q$, rather than polynomially in $q$. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products $F\wr S_n$, whose cost depends on the cost of a QFT over $F$ and the size of its representation registers. This removes the restriction $|F|=\operatorname{poly}(n)$ required by previous generic constructions and can yield exponential improvements when $F$ itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.

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arXiv2026

Optimal cloning of mixed states

We consider the problem of approximate cloning of quantum states: given $n$ copies of an unknown state $\rho \in \mathbb{C}^{d \times d}$, prepare an $(n+k)$-copy state with high fidelity to $\rho^{\otimes (n+k)}$. Werner's pure state cloner is the optimal channel for the pure state case, and shows that $n = \Theta(kd/\varepsilon)$ copies are necessary and sufficient to clone $k$ additional copies of an unknown pure state to fidelity $1-\varepsilon$. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given $n$ copies of a mixed state, randomly purify your input, apply Werner's channel in the larger Hilbert space, and then trace out the auxiliary registers. This gives a mixed state cloner using $n = O(krd/\varepsilon)$ copies to clone rank-$r$ states. Can one do any better? We show that the answer is no: one must use $n = \Omega(krd/\varepsilon)$ copies. We prove our lower bound by studying the special case of projector cloning, in which the input state $\rho$ is promised to be of the form $P/r$, where $P$ is a rank-$r$ orthogonal projector. As a further application of our techniques, we consider the closely related problem of approximate transposition of quantum states, where one seeks to convert $\rho^{\otimes n}$ to a $k$-copy state with high fidelity to $(\rho^T)^{\otimes k}$. Here, we again show $n = \Theta(krd/\varepsilon)$ copies are necessary and sufficient for this task.

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arXiv2026

Sample complexity of quantum resource testing via one-shot quantum blurring

Quantum resource testing is a fundamental primitive of quantum information processing, profoundly connected to resource manipulation. Its goal is to discriminate $n$ copies of a given resourceful state $\rho$ from all free (i.e., resourceless) states; key instances for applications are entanglement testing and quantum magic testing. The asymptotic characterisation relies on the recently proven Generalised Quantum Stein's Lemma (GQSL), which establishes the rate of decay of the false negative error probability for a fixed false positive error probability. This result, however, is intrinsically asymptotic and thus can provide no finite-resource guarantees, which makes its practical implications unclear. Here, we establish the first rigorous finite-$n$ bounds on quantum resource testing and hence quantum resource manipulation, thus strengthening the GQSL and providing explicit estimates on the number of copies needed to achieve a prescribed performance. As notable consequences, we obtain (a) the convergence of the regularised R\'enyi relative entropies of a resource, which settles the important open problem from [Fang/Hayashi, arXiv:2508.12901, IEEE ToIT 72:6, 2026]; and (b) the first sample-complexity bound for asymmetric resource testing: for any fixed false positive error probability, a false negative error probability of at most $\delta$ can be achieved with $n=O\left(\frac{\log(1/\delta)}{D^\infty(\rho\|F)}\right)$ copies of $\rho$, in the limit where $\delta \to 0$.

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arXiv2026

Classical shadows with arbitrary group representations

Classical shadows (CS) has recently emerged as an important framework to efficiently predict properties of an unknown quantum state. A common strategy in CS protocols is to parametrize the basis in which one measures the state by a random group action; many examples of this have been proposed and studied on a case-by-case basis. In this work, we present a unified theory that allows us to simultaneously understand CS protocols based on sampling from general group representations, extending previous approaches that worked in simplified (multiplicity-free) settings. We identify a class of measurement bases which we call "centralizing bases" that allows us to analytically characterize and invert the measurement channel, minimizing classical post-processing costs. We complement this analysis by deriving general bounds on the sample-complexity necessary to obtain estimates of a given precision. Beyond its unification of previous CS protocols, our method allows us to readily generate new protocols based on other groups, or different representations of previously considered ones. For example, we characterize novel shadow protocols based on sampling from the spin and tensor representations of $\textsf{SU}(2)$, symmetric and orthogonal groups, and the exceptional Lie group $G_2$.

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arXiv2026

Optimal pure state cloning and transposition are complementary channels

State cloning and state transposition are fundamental transformations which, despite being desirable, cannot be perfectly realised due to two conceptually distinct constraints of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, despite these different constraints, the best physically allowed realisation of both transformations arises from a single physical process described by an isometry, which simultaneously implements their best possible approximations. We first determine the optimal fidelity for transforming $N$ qudits into $K$ copies of their transposition and show that, for pure input states, it is achieved by an estimation strategy, which is the unique optimal strategy under the worst-case fidelity figure of merit. We further prove that the corresponding $N \to K$ transposition map is the complementary channel of the optimal universal symmetric $N \to N + K$ quantum cloning machine on pure states. We then present an explicit quantum circuit that realises $N \to K$ transposition and $N \to N + K$ cloning in parallel and analyse its gate efficiency. Finally, we investigate mixed-state $N \to 1$ qudit transposition and determine its maximal performance in terms of white-noise visibility, yielding the structural physical approximation of transposition in the multicopy regime.

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arXiv2026

Stronger Welch Bounds and Optimal Approximate $k$-Designs

A fundamental question asks how uniformly finite sets of pure quantum states can be distributed in a Hilbert space. The Welch bounds address this question, and are saturated by $k$-designs, i.e. sets of states reproducing the $k$-th Haar moments. However, these bounds quickly become uninformative when the number of states is below that required for an exact $k$-design. We derive strengthened Welch-type inequalities that remain sharp in this regime by exploiting rank constraints from partial transposition and spectral properties of the partially transposed Haar moment operator. We prove that the deviation from the Welch bound captures the average-case approximation error, hence characterizing a natural notion of minimum achievable error at fixed cardinality. For $k=3$, we prove that SICs and complete MUB sets saturate our bounds, making them optimal approximate 3-designs of their cardinality. This leads to a natural variational criterion to rule out the existence of a complete set MUBs, which we use to obtain numerical evidence against such set in dimension $6$. As a key technical ingredient, we compute the complete spectrum of the partially transposed symmetric-subspace projector, including multiplicities and eigenvectors, which may find applications beyond the present work.

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arXiv2025

Detection of many-body entanglement partitions in a quantum computer

We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, $m$-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.

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arXiv2025

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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arXiv2025

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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arXiv2025

High-dimensional quantum Schur transforms

The quantum Schur transform has become a foundational quantum algorithm, yet even after two decades since the seminal 2005 paper by Bacon, Chuang, and Harrow (BCH), some aspects of the transform remain insufficiently understood. Moreover, an alternative approach proposed by Krovi in 2018 was recently found to contain a crucial error. In this paper, we present a corrected version of Krovi's algorithm along with a detailed treatment of the high-dimensional version of the BCH Schur transform. This high-dimensional focus makes the two versions of the transform practical for regimes where the number of qudits $n$ is smaller than the local dimension $d$, with Krovi's algorithm scaling as $\widetilde{O}(n^4)$ and BCH as $\widetilde{O}(\min(n^5,nd^4))$. Our work addresses a key gap in the literature, strengthening the algorithmic foundations of a wide range of results that rely on Schur--Weyl duality in quantum information theory and quantum computation.

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arXiv2025

Nearly optimal algorithms to learn sparse quantum Hamiltonians in physically motivated distances

We study the problem of learning Hamiltonians $H$ that are $s$-sparse in the Pauli basis, given access to their time evolution. Although Hamiltonian learning has been extensively investigated, two issues recur in much of the existing literature: the absence of matching lower bounds and the use of mathematically convenient but physically opaque error measures. We address both challenges by introducing two physically motivated distances between Hamiltonians and designing a nearly optimal algorithm with respect to one of these metrics. The first, time-constrained distance, quantifies distinguishability through dynamical evolution up to a bounded time. The second, temperature-constrained distance, captures distinguishability through thermal states at bounded inverse temperatures. We show that $s$-sparse Hamiltonians with bounded operator norm can be learned in both distances with $O(s \log(1/\epsilon))$ experiments and $O(s^2/\epsilon)$ evolution time. For the time-constrained distance, we further establish lower bounds of $\Omega((s/n)\log(1/\epsilon) + s)$ experiments and $\Omega(\sqrt{s}/\epsilon)$ evolution time, demonstrating near-optimality in the number of experiments. As an intermediate result, we obtain an algorithm that learns every Pauli coefficient of $s$-sparse Hamiltonians up to error $\epsilon$ in $O(s\log(1/\epsilon))$ experiments and $O(s/\epsilon)$ evolution time, improving upon several recent results. The source of this improvement is a new isolation technique, inspired by the Valiant-Vazirani theorem (STOC'85), which shows that NP is as easy as detecting unique solutions. This isolation technique allows us to query the time evolution of a single Pauli coefficient of a sparse Hamiltonian--even when the Pauli support of the Hamiltonian is unknown--ultimately enabling us to recover the Pauli support itself.

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arXiv2025

Entanglement recycling in two-step port-based teleportation

A protocol involving the repetitive (twofold, to be precise) application of PBT protocol to the same resource is studied. The quantities characterizing the resulting protocol, so-called \textit{two-step PBT}, namely \textit{enatnglement fidelity} and \textit{success probability} are provided for two scenarios, relying on application of pretty-good measurement, i.e. deterministic and probabilistic PBT with non-EPR resource. This results show that two-step PBT is an accurate protocol, provided the resource is sufficiently large. In particular, the deterministic two-step PBT obtains fidelity that is remarkably close to the optimal MPBT fidelity for teleportation of two quantum states. Additionally, the \textit{recycling fidelity}, i.e. the quantity characterizing the degradation of the resource state is calculated for repetitive application of probabilistic protocol, for both EPR and optimized resource, showing that entanglement recycling with two-step PBT is possible in the former case as well.

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arXiv2025

Group-Adapted Irreducible Matrix Units for the Walled Brauer Algebra

This paper investigates the representation theory of the algebra of partially transposed permutation operators, $\mathcal{A}^d_{p,p}$, which provides a matrix representation for the abstract walled Brauer algebra. This algebra has recently gained significant attention due to its relevance in quantum information theory, particularly in the efficient quantum circuit implementation of the mixed Schur-Weyl transform. In contrast to previous Gelfand-Tsetlin type approaches, our main technical contribution is the explicit construction of irreducible matrix units in the second-highest ideal that are group-adapted to the action of $\mathbb{C}[S_p]\times \mathbb{C}[S_p]$ subalgebra, where $S_p$ is the symmetric group. This approach suggests a recursive method for constructing irreducible matrix units in the remaining ideals of the algebra. The framework is general and applies to systems with arbitrary numbers of components and local dimensions. In addition, we present a complementary construction method based on tensor networks of Clebsch-Gordan coefficients of the unitary group. This approach enables the construction of all group-adapted irreducible matrix units, but requires knowledge of certain Littlewood-Richardson coefficients. This method can be successfully applied for a reasonably small number of particles with the support of dedicated software. The obtained results are applied to a special class of operators motivated by the mathematical formalism appearing in all variants of the port-based teleportation protocols through the mixed Schur-Weyl duality. We demonstrate that the given irreducible matrix units are, in fact, eigenoperators for the considered class.

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arXiv2024

Quantum Programmable Reflections

Similar to a classical processor, which is an algorithm for reading a program and executing its instructions on input data, a universal programmable quantum processor is a fixed quantum channel that reads a quantum program $\lvert\psi_{U}\rangle$ that causes the processor to approximately apply an arbitrary unitary $U$ to a quantum data register. The present work focuses on a class of simple programmable quantum processors for implementing reflection operators, i.e. $U = e^{i \pi \lvert\psi\rangle\langle\psi\rvert}$ for an arbitrary pure state $\lvert\psi\rangle$ of finite dimension $d$. Unlike quantum programs that assume query access to $U$, our program takes the form of independent copies of the state to be reflected about $\lvert\psi_U\rangle = \lvert\psi\rangle^{\otimes n}$. We then identify the worst-case optimal algorithm among all processors of the form $\text{tr}_{\text{Program}}[V (\lvert\phi\rangle\langle\phi\rvert \otimes (\lvert\psi\rangle\langle\psi\rvert)^{\otimes n}) V^\dagger]$ where the algorithm $V$ is a unitary linear combination of permutations. By generalizing these algorithms to processors for arbitrary-angle rotations, $e^{i \alpha \lvert\psi\rangle\langle\psi\rvert}$ for $\alpha \in \mathbb R$, we give a construction for a universal programmable processor with better scaling in $d$. For programming reflections, we obtain a tight analytical lower bound on the program dimension by bounding the Holevo information of an ensemble of reflections applied to an entangled probe state. The lower bound makes use of a block decomposition of the uniform ensemble of reflected states with respect to irreps of the partially transposed permutation matrix algebra, and two representation-theoretic conjectures based on extensive numerical evidence.

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arXiv2024

Bosonic randomized benchmarking with passive transformations

Randomized benchmarking (RB) is the most commonly employed protocol for the characterization of unitary operations in quantum circuits due to its reasonable experimental requirements and robustness against state preparation and measurement (SPAM) errors. So far, the protocol has been limited to discrete or fermionic systems, whereas extensions to bosonic systems have been unclear for a long time due to challenges arising from the underlying infinite-dimensional Hilbert spaces. In this work, we close the gap for bosonic systems and develop an RB protocol to benchmark passive Gaussian transformations on any particle number subspace, which we call bosonic passive RB. The protocol is based on the recently developed filtered RB framework and is designed to isolate the multitude of exponential decays arising for passive bosonic transformations. We give explicit formulas and a Julia implementation for the necessary post-processing of the experimental data. We also analyze the sampling complexity of bosonic passive RB by deriving analytical expressions for the variance. They show a mild scaling with the number of modes, suggesting that passive bosonic RB is experimentally feasible for a moderate number of modes. We focus on experimental settings involving Fock states and particle number resolving measurements, but also discuss Gaussian settings, deriving first results for heterodyne measurements.

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arXiv2024

On compatibility of binary qubit measurements

Deciding which sets of quantum measurements allow a simultaneous readout is a central problem in quantum measurement theory. The problem is relevant not only from the foundational perspective but also has direct applications in quantum correlation problems fueled by incompatible measurements. Although central, only a few analytical criteria exist for deciding the incompatibility of general sets of measurements. This work approaches the problem through functions defined on the Boolean hypercube and their Fourier transformations. We show that this reformulation of the problem leads to a complete geometric characterisation of joint measurability of any finite set of unbiased binary qubit measurements and gives a necessary condition for the biased case. We discuss our results in the realm of quantum steering, where they translate into a family of steering inequalities. When certain unbiasedness conditions are fulfilled, these criteria are tight, hence fully characterizing the steering problem when the trusted party holds a qubit, and the untrusted party performs any finite number of binary measurements. We further discuss how our results point towards a second-order cone programming approach to measurement incompatibility and compare this to the predominantly used semi-definite programming-based techniques. We use our approach to falsify an existing conjecture on measurement incompatibility of special sets of measurements.

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arXiv2024

Permutation tests for quantum state identity

The quantum analogue of the equality function, known as the quantum state identity problem, is the task of deciding whether $n$ unknown quantum states are equal or unequal, given the promise that all states are either pairwise orthogonal or identical. Under the one-sided error requirement, it is known that the permutation test is optimal for this task, and for two input states this coincides with the well-known Swap test. Until now, the optimal measurement in the general two-sided error regime was unknown. Under more specific promises, the problem can be solved approximately or even optimally with simpler tests, such as the circle test. This work attempts to capture the underlying structure of the quantum state identity problem. Using tools from semidefinite programming and representation theory, we (i) give an optimal test for any input distribution without the one-sided error requirement by writing the problem as an SDP, giving the exact solutions to the primal and dual programs and showing that the two values coincide; (ii) propose a general $G$-test which uses an arbitrary subgroup $G$ of $\text{S}_n$, giving an analytic expression of the performance of the specific test, and (iii) give an approximation of the permutation test using only a classical permutation and $n-1$ Swap tests.

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