arXiv subjects

Piotr Dulian

Publications and source records attributed to Piotr Dulian.

6 recordsLinked to original sources

arXiv2026

Quantum Circuit Overhead

We introduce a measure for evaluating the efficiency of finite universal quantum gate sets $\mathcal{S}$, called the Quantum Circuit Overhead (QCO), and the related notion of $T$-Quantum Circuit Overhead ($T$-QCO). QCO compares the circuit length required by $\mathcal S$ with the best possible length among gate sets of the same size. The $T$-QCO adapts this idea to cost models in which only selected costly gates are counted, while cheap operations are absorbed into an effective gate set. We demonstrate the usefulness of the ($T$-)QCO by extensive numerical calculations of its upper bounds, providing insight into the efficiency of various choices of single-qubit $\mathcal{S}$, including Haar-random gate sets and the gate sets derived from finite subgroups, such as Clifford and Hurwitz groups. In particular, our results suggest that, in terms of the upper bounds on the $T$-QCO, the famous T gate is a highly non-optimal choice for the completion of the Clifford gate set, even among the gates of order 8. We identify the optimal choices of such completions for both finite subgroups.

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arXiv2026

QMetro++ -- Python optimization package for large scale quantum metrology with customized strategy structures

QMetro++ is a Python package that provides a set of tools for identifying optimal estimation protocols that maximize quantum Fisher information (QFI). Optimization can be performed for arbitrary configurations of input states, parameter-encoding channels, noise correlations, control operations, and measurements. The use of tensor networks and an iterative see-saw algorithm allows for an efficient optimization even in the regime of a large number of channel uses ($N\approx100$). Additionally, the package includes implementations of the recently developed methods for computing fundamental upper bounds on QFI, which serve as benchmarks for assessing the optimality of numerical optimization results. All functionalities are wrapped up in a user-friendly interface which enables the definition of strategies at various levels of detail.

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arXiv2025

Symplectic Structures in Quantum Entanglement

In this work, we explore the implications of applying the formalism of symplectic geometry to quantum mechanics, particularly focusing on many-particle systems. We extend the concept of a symplectic indicator of entanglement, originally introduced by Sawicki et al. \cite{sawicki2011}, to these complex systems. Specifically, we demonstrate that the restriction of the symplectic structure to manifolds comprising all states characterized by isospectral reduced one-particle density matrices, $ M_{μ(ψ)}^0 $, exhibits degeneracy for non-separable states. We prove that the degree of degeneracy at any given state $ \ketφ \in M_{μ(ψ)}^0 $ corresponds to the degree of degeneracy of the symplectic form $ ω$ when restricted to the manifold of states that are locally unitary equivalent with $ \ketφ $. Additionally, we provide a physical interpretation of this symplectic indicator of entanglement, articulating it as an inherent ambiguity within the associated classical dynamical framework. Our findings underscore the pivotal role of symplectic geometry in elucidating entanglement properties in quantum mechanics and suggest avenues for further exploration into the geometric structures underlying quantum state spaces.

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arXiv2025

Quantum metrology using quantum combs and tensor network formalism

We develop an efficient algorithm for determining optimal adaptive quantum estimation protocols with arbitrary quantum control operations between subsequent uses of a probed channel. We introduce a tensor network representation of an estimation strategy, which drastically reduces the time and memory consumption of the algorithm, and allows us to analyze metrological protocols involving up to $N=50$ qubit channel uses, whereas the state-of-the-art approaches are limited to $N<5$. The method is applied to study the performance of the optimal adaptive metrological protocols in presence of various noise types, including correlated noise.

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arXiv2024

A random matrix model for random approximate $t$-designs

For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider the uniform measure $ν_\mathcal{S}$ whose support is given by $\mathcal{S}$. The measure $ν_\mathcal{S}$ can be regarded as a $δ(ν_\mathcal{S},t)$-approximate $t$-design, $t\in\mathbb{Z}_+$. We propose a random matrix model that aims to describe the probability distribution of $δ(ν_\mathcal{S},t)$ for any $t$. Our model is given by a block diagonal matrix whose blocks are independent, given by Gaussian or Ginibre ensembles, and their number, size and type is determined by $t$. We prove that, the operator norm of this matrix, $δ({t})$, is the random variable to which $\sqrt{|\mathcal{S}|}δ(ν_\mathcal{S},t)$ converges in distribution when the number of elements in $\mathcal{S}$ grows to infinity. Moreover, we characterize our model giving explicit bounds on the tail probabilities $\mathbb{P}(δ(t)>2+ε)$, for any $ε>0$. We also show that our model satisfies the so-called spectral gap conjecture, i.e. we prove that with the probability $1$ there is $t\in\mathbb{Z}_+$ such that $\sup_{k\in\mathbb{Z}_{+}}δ(k)=δ(t)$. Numerical simulations give convincing evidence that the proposed model is actually almost exact for any cardinality of $\mathcal{S}$. The heuristic explanation of this phenomenon, that we provide, leads us to conjecture that the tail probabilities $\mathbb{P}(\sqrt{\mathcal{S}}δ(ν_\mathcal{S},t)>2+ε)$ are bounded from above by the tail probabilities $\mathbb{P}(δ(t)>2+ε)$ of our random matrix model. In particular our conjecture implies that a Haar random set $\mathcal{S}\subset U(d)$ satisfies the spectral gap conjecture with the probability $1$.

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arXiv2023

Matrix concentration inequalities and efficiency of random universal sets of quantum gates

For a random set $\mathcal{S} \subset U(d)$ of quantum gates we provide bounds on the probability that $\mathcal{S}$ forms a $δ$-approximate $t$-design. In particular we have found that for $\mathcal{S}$ drawn from an exact $t$-design the probability that it forms a $δ$-approximate $t$-design satisfies the inequality $\mathbb{P}\left(δ\geq x \right)\leq 2D_t \, \frac{e^{-|\mathcal{S}| x \, \mathrm{arctanh}(x)}}{(1-x^2)^{|\mathcal{S}|/2}} = O\left( 2D_t \left( \frac{e^{-x^2}}{\sqrt{1-x^2}} \right)^{|\mathcal{S}|} \right)$, where $D_t$ is a sum over dimensions of unique irreducible representations appearing in the decomposition of $U \mapsto U^{\otimes t}\otimes \bar{U}^{\otimes t}$. We use our results to show that to obtain a $δ$-approximate $t$-design with probability $P$ one needs $O( δ^{-2}(t\log(d)-\log(1-P)))$ many random gates. We also analyze how $δ$ concentrates around its expected value $\mathbb{E}δ$ for random $\mathcal{S}$. Our results are valid for both symmetric and non-symmetric sets of gates.

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