arXiv subjects

Julien Zylberman

Publications and source records attributed to Julien Zylberman.

11 recordsLinked to original sources

arXiv2026

Logarithmic-depth quantum state preparation of polynomials

Quantum state preparation is a central primitive in many quantum algorithms, yet it is generally resource intensive, with efficient constructions known only for structured families of states. This work introduces a method for preparing quantum states whose amplitudes are given by a degree$-d$ polynomial, using circuits with logarithmic depth in the number $n$ of qubits and only $\mathcal O(n)$ ancilla qubits, improving previous approaches that required linear-depth circuits. The construction first relies on a block-encoding of an affine diagonal operator based on its Pauli-basis decomposition, which involves only $n$ terms. A modified linear-combination-of-unitaries (LCU) technique is introduced to implement this decomposition in logarithmic depth, together with a novel circuit for the EXACT-one oracle that flags basis states in which exactly one qubit is in the state $|1\rangle$. It then uses a generalized quantum eigenvalue transformation (GQET) to promote this affine operator to an arbitrary degree polynomial. Theoretical analysis and numerical simulations are reported along with a proof-of-principle implementation on a trapped-ion quantum processor using $14$ qubits and more than $500$ primitive quantum gates. Because polynomial approximations are ubiquitous in scientific computing, this construction provides a scalable and resource-efficient approach to quantum state preparation, further improving the potential of quantum algorithms in fields such as chemistry, physics, engineering, and finance.

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arXiv2026

Quantum algorithm for anisotropic diffusion and convection equations with vector norm scaling

In this work, we tackle the resolution of partial differential equations (PDEs) on digital quantum computers. Two fundamental PDEs are addressed: the anisotropic diffusion equation and the anisotropic convection equation. We present a quantum numerical scheme consisting of three steps: quantum state preparation, evolution with diagonal operators, and measurement of observables of interest. The evolution step relies on a high-order centered finite difference and a product formula approximation, also known as Trotterization. We provide novel vector-norm analysis to bound the different sources of error. We prove that the number of time-steps required in the evolution can be reduced by a factor $\Theta (16^n)$ for the diffusion equation, and $\Theta (4^n)$ for the convection equation, where $n$ is the number of qubits per dimension, an exponential reduction compared to the previously established operator-norm analysis.

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arXiv2025

Trotter-based quantum algorithm for solving transport equations with exponentially fewer time-steps

The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the multidimensional transport equation with space- and time-dependent coefficients. We present a quantum numerical scheme based on three steps: quantum state preparation, evolution, and measurement of relevant observables. The evolution step combines a high-order centered finite difference with a time-splitting scheme based on product formula approximations, also known as Trotterization. We introduce novel vector-norm analysis and prove that the number of time-steps can be reduced by a factor exponential in the number of qubits compared to previously established operator-norm analysis, thereby significantly lowering the projected computational resources. We also present efficient quantum circuits and numerical simulations that confirm the predicted vector-norm scaling. We report results on real quantum hardware for the one-dimensional convection equation, and solve a non-linear ordinary differential equation via its associated Liouville equation, a particular case of transport equations. This work provides a practical framework for efficiently simulating transport phenomena on quantum computers, with potential applications in plasma physics, molecular gas dynamics and non-linear dynamical systems, including chaotic systems.

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arXiv2025

Real-Space Chemistry on Quantum Computers: A Fault-Tolerant Algorithm with Adaptive Grids and Transcorrelated Extension

First-quantized, real-space formulations of quantum chemistry on quantum computers are appealing: qubit count scales logarithmically with spatial resolution, and Coulomb operators achieve quadratic instead of quartic computational scaling of two-electron interactions. However, existing schemes employ uniform discretizations, so the resolution required to capture electron-nuclear cusps in high-density regions oversamples low-density regions, wasting computational resources. We address this by deploying non-uniform, molecule-adaptive grids that concentrate points where electronic density is high. Using Voronoi partitions of these grids, the molecular Hamiltonian is expressed in a Hermitian form and in a transcorrelated, isospectral form that eliminates Coulomb singularities and yields cusp-free eigenfunctions. Both formulations slot naturally into quantum eigenvalue solvers: Hermitian Quantum Phase Estimation (QPE) and the recent generalised Quantum Eigenvalue Estimation (QEVE) protocol for its non-Hermitian, transcorrelated counterpart. Numerical validation on benchmark systems confirms that this non-heuristic ab initio framework offers a promising path for accurate ground-state chemistry on quantum hardware.

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arXiv2024

Fast Laplace transforms on quantum computers

While many classical algorithms rely on Laplace transforms, it has remained an open question whether these operations could be implemented efficiently on quantum computers. In this work, we introduce the Quantum Laplace Transform (QLT), which enables the implementation of $N\times N$ discrete Laplace transforms on quantum states encoded in $\lceil \log_2(N)\rceil$-qubits. In many cases, the associated quantum circuits have a depth that scales with $N$ as $O(\log(\log(N)))$ and a size that scales as $O(\log(N))$, requiring exponentially fewer operations and double-exponentially less computational time than their classical counterparts. These efficient scalings open the possibility of developing a new class of quantum algorithms based on Laplace transforms, with potential applications in physics, engineering, chemistry, machine learning, and finance.

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arXiv2024

Efficient Quantum Circuits for Non-Unitary and Unitary Diagonal Operators with Space-Time-Accuracy trade-offs

Unitary and non-unitary diagonal operators are fundamental building blocks in quantum algorithms with applications in the resolution of partial differential equations, Hamiltonian simulations, the loading of classical data on quantum computers (quantum state preparation) and many others. In this paper, we introduce a general approach to implement unitary and non-unitary diagonal operators with efficient-adjustable-depth quantum circuits. The depth, i.e., the number of layers of quantum gates of the quantum circuit, is reducible with respect either to the width, i.e, the number of ancilla qubits, or to the accuracy between the implemented operator and the target one. While exact methods have an optimal exponential scaling either in terms of size, i.e., the total number of primitive quantum gates, or width, approximate methods prove to be efficient for the class of diagonal operators depending on smooth, at least differentiable, functions. Our approach is general enough to allow any method for diagonal operators to become adjustable-depth or approximate, decreasing the depth of the circuit by increasing its width or its approximation level. This feature offers flexibility and can match with the hardware limitations in coherence time or cumulative gate error. We illustrate these methods by performing quantum state preparation and non-unitary-real-space simulation of the diffusion equation. This simulation paves the way to efficient implementations of stochastic models useful in physics, chemistry, biology, image processing and finance.

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arXiv2023

Polylogarithmic-depth controlled-NOT gates without ancilla qubits

Controlled operations are fundamental building blocks of quantum algorithms. Decomposing $n$-control-NOT gates ($C^n(X)$) into arbitrary single-qubit and CNOT gates, is a crucial but non-trivial task. This study introduces $C^n(X)$ circuits outperforming previous methods in the asymptotic and non-asymptotic regimes. Three distinct decompositions are presented: an exact one using one borrowed ancilla with a circuit depth $\Theta\left(\log(n)^{3}\right)$, an approximating one without ancilla qubits with a circuit depth $\mathcal O \left(\log(n)^{3}\log(1/\epsilon)\right)$ and an exact one with an adjustable-depth circuit which decreases with the number $m\leq n$ of ancilla qubits available as $O(log(2n/m)^3+log(m/2))$. The resulting exponential speedup is likely to have a substantial impact on fault-tolerant quantum computing by improving the complexities of countless quantum algorithms with applications ranging from quantum chemistry to physics, finance and quantum machine learning.

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arXiv2023

Efficient Quantum State Preparation with Walsh Series

A new approximate Quantum State Preparation (QSP) method is introduced, called the Walsh Series Loader (WSL). The WSL approximates quantum states defined by real-valued functions of single real variables with a depth independent of the number $n$ of qubits. Two approaches are presented: the first one approximates the target quantum state by a Walsh Series truncated at order $O(1/\sqrtε)$, where $ε$ is the precision of the approximation in terms of infidelity. The circuit depth is also $O(1/\sqrtε)$, the size is $O(n+1/\sqrtε)$ and only one ancilla qubit is needed. The second method represents accurately quantum states with sparse Walsh series. The WSL loads $s$-sparse Walsh Series into $n$-qubits with a depth doubly-sparse in $s$ and $k$, the maximum number of bits with value $1$ in the binary decomposition of the Walsh function indices. The associated quantum circuit approximates the sparse Walsh Series up to an error $ε$ with a depth $O(sk)$, a size $O(n+sk)$ and one ancilla qubit. In both cases, the protocol is a Repeat-Until-Success (RUS) procedure with a probability of success $P=Θ(ε)$, giving an averaged total time of $O(1/ε^{3/2})$ for the WSL (resp. $O(sk/ε)$ for the sparse WSL). Amplitude amplification can be used to reduce by a factor $O(1/\sqrtε)$ the total time dependency with $ε$ but increases the size and depth of the associated quantum circuits, making them linearly dependent on $n$. These protocols give overall efficient algorithms with no exponential scaling in any parameter. They can be generalized to any complex-valued, multi-variate, almost-everywhere-differentiable function. The Repeat-Until-Success Walsh Series Loader is so far the only method which prepares a quantum state with a circuit depth and an averaged total time independent of the number of qubits.

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arXiv2022

Quantum circuits for discrete-time quantum walks with position-dependent coin operator

The aim of this paper is to build quantum circuits that implement discrete-time quantum walks having an arbitrary position-dependent coin operator. The position of the walker is encoded in base 2: with $n$ wires, each corresponding to one qubit, we encode $2^n$ position states. The data necessary to define an arbitrary position-dependent coin operator is therefore exponential in $n$. We first propose a circuit implementing the position-dependent coin operator, that is naive, in the sense that it has exponential depth and implements sequentially all appropriate position-dependent coin operators. We then propose a circuit that "transfers" all the depth into ancillae, yielding a final depth that is linear in $n$ at the cost of an exponential number of ancillae. The main idea of this linear-depth circuit is to implement in parallel all coin operators at the different positions. Finally, we extend the result of Ref. [2] from position-dependent unitaries which are diagonal in the position basis to position-dependent $2 \times 2$-block-diagonal unitaries: indeed, we show that for a position dependence of the coin operator (the block-diagonal unitary) which is smooth enough, one can find an efficient quantum-circuit implementation approximating the coin operator up to an error $\epsilon$ (in terms of the spectral norm), the depth and size of which scale as $O(1/\epsilon)$. A typical application of the efficient implementation would be the quantum simulation of a relativistic spin-1/2 particle on a lattice, coupled to a smooth external gauge field; notice that recently, quantum spatial-search schemes have been developed which use gauge fields as the oracle, to mark the vertex to be found [3, 4]. A typical application of the linear-depth circuit would be when there is spatial noise on the coin operator (and hence a non-smooth dependence in the position).

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arXiv2022

Quantum spatial search with electric potential : long-time dynamics and robustness to noise

We present various results on the scheme introduced , which is a quantum spatial-search algorithm on a two-dimensional (2D) square spatial grid, realized with a 2D Dirac discrete-time quantum walk (DQW) coupled to a Coulomb electric field centered on the marked node. In such a walk, the electric term acts as the oracle of the algorithm, and the free walk (i.e., without electric term) acts as the "diffusion" part, as it is called in Grover's algorithm. The results are the following. First, we run simulations of this electric Dirac DQW during longer times than explored in Ref.\ \cite{ZD21}, and observe that there is a second localization peak around the node marked by the oracle, reached in a time $O(\sqrt{N})$, where $N$ is the number of nodes of the 2D grid, with a localization probability scaling as $O(1/\ln N)$. This matches the state-of-the-art 2D DQW search algorithms before amplitude amplification. We then study the effect of adding noise on the Coulomb potential, and observe that the walk, especially the second localization peak, is highly robust to spatial noise, more modestly robust to spatiotemporal noise, and that the first localization peak is even highly robust to spatiotemporal noise.

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arXiv2022

Hybrid Quantum-Classical Algorithm for Hydrodynamics

A new model of nonlinear charged quantum relativistic fluids is presented. This model can be discretized into Discrete Time Quantum Walks (DTQWs), and a new hybrid (quantum-classical) algorithm for implementing these walks on NISQ devices is proposed. High resolution (up to $N=2^{17}$ grid points) hybrid numerical simulations of relativistic and non-relativistic hydrodynamical shocks on current IBM NISQs are performed with this algorithm and shown to reproduce equivalent simulations on classical computers. This work demonstrates that nonlinear fluid dynamics can be simulated on NISQs, and opens the door to simulating other, quantum and non-quantum fluids, including plasmas, with more general quantum walks and quantum automata.

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