Searcharxiv⌕ Search

arXiv subjects

Rahul Sarkar

Publications and source records attributed to Rahul Sarkar.

14 recordsLinked to original sources
arXiv2026

Constrained minimax approximation for quantum signal processing

Quantum signal processing (QSP) provides a simple and efficient framework for implementing polynomial transformations using quantum circuits. Its classical design stage leads to a constrained minimax approximation problem: find a polynomial of prescribed parity that approximates a target function uniformly on a fitting set while remaining bounded in magnitude by one on the domain $[0,1]$, which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.

quant-ph↗
arXiv2026

Quantum element-wise transforms

Quantum algorithms for basic numerical linear algebraic tasks have proven essential for translating diverse problems to a unified quantum computational context. Many of these tasks -- e.g., applying a polynomial function to the spectrum of a matrix embedded in a unitary process (a so-called block encoding), or taking linear combinations of block encodings -- are well-addressed by techniques like quantum singular value transformation (QSVT) or linear combination of unitaries (LCU). However, there exist useful matrix transforms whose realization by existing quantum algorithms is unclear or inefficient. In this work we construct improved quantum algorithms for some of these transforms, the simplest of which is a polynomial function applied element-wise. We show the space required to compute quantum element-wise transforms can be reduced exponentially in the degree of the applied function compared to prior work, and raise and rectify errors in previous constructions. We present our algorithms alongside applications to machine learning, simulation, and signal processing.

quant-ph↗
arXiv2026

Overlapped groupings for quantum energy estimation: Maximal variance reduction and deterministic algorithms for reducing variance

Grouping-based measurement strategies are widely used to reduce measurement complexity in near-term quantum algorithms. While these schemes have typically produced disjoint groups, recently this has been relaxed in what is known as overlapped grouping or coefficient splitting where operators may appear in more than one compatible group. In recent work, it has been numerically shown that this strategy can reduce the variance of energy estimates on small benchmark problems, motivating both the application and further analysis of the method. Here we prove that overlapped grouping for energy estimation can lead to a maximal variance reduction that is linear in the number of Hamiltonian terms. We introduce a new algorithm which we call repacking to transform existing groups into overlapped groups, and we show this repacking procedure iteratively reduces variance under mild assumptions. We also perform numerical simulations with Hamiltonians up to $44$ qubits and $575 \cdot 10^{3}$ terms, assessing overlapped grouping at scale on problems of practical importance. Our numerics show that the variance reduction relative to state-of-the-art (disjoint) grouping increases linearly with the problem size, suggesting that overlapped grouping methods can be a powerful strategy for quantum energy estimation at the scale of Megaquop computers and beyond.

quant-ph↗
arXiv2025

Inverse nonlinear fast Fourier transform on SU(2) with applications to quantum signal processing

The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on $\mathrm{SU}(1,1)$ has been widely studied, its $\mathrm{SU}(2)$ variant has only recently attracted attention due to emerging applications in quantum signal processing (QSP) and quantum singular value transformation (QSVT). In this paper, we investigate the inverse NLFT on $\mathrm{SU}(2)$ and establish the numerical stability of the layer stripping algorithm for the first time under suitable conditions. Furthermore, we develop a fast and numerically stable algorithm, called inverse nonlinear fast Fourier transform, for performing inverse NLFT with near-linear complexity. This algorithm is applicable to computing phase factors for both QSP and the generalized QSP (GQSP).

quant-ph↗
arXiv2024

Time-dependent Hamiltonian Simulation via Magnus Expansion: Algorithm and Superconvergence

Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a new time-dependent Hamiltonian simulation algorithm based on the Magnus series expansion that exhibits both features. Importantly, when applied to unbounded Hamiltonian simulation in the interaction picture, we prove that the commutator in the second-order algorithm leads to a surprising fourth-order superconvergence, with an error preconstant independent of the number of spatial grids. This extends the qHOP algorithm [An, Fang, Lin, Quantum 2022] based on first-order Magnus expansion, and the proof of superconvergence is based on semiclassical analysis that is of independent interest.

quant-ph↗
arXiv2024

Uniform $\mathcal{C}^k$ Approximation of $G$-Invariant and Antisymmetric Functions, Embedding Dimensions, and Polynomial Representations

For any subgroup $G$ of the symmetric group $\mathcal{S}_n$ on $n$ symbols, we present results for the uniform $\mathcal{C}^k$ approximation of $G$-invariant functions by $G$-invariant polynomials. For the case of totally symmetric functions ($G = \mathcal{S}_n$), we show that this gives rise to the sum-decomposition Deep Sets ansatz of Zaheer et al. (2018), where both the inner and outer functions can be chosen to be smooth, and moreover, the inner function can be chosen to be independent of the target function being approximated. In particular, we show that the embedding dimension required is independent of the regularity of the target function, the accuracy of the desired approximation, as well as $k$. Next, we show that a similar procedure allows us to obtain a uniform $\mathcal{C}^k$ approximation of antisymmetric functions as a sum of $K$ terms, where each term is a product of a smooth totally symmetric function and a smooth antisymmetric homogeneous polynomial of degree at most $\binom{n}{2}$. We also provide upper and lower bounds on $K$ and show that $K$ is independent of the regularity of the target function, the desired approximation accuracy, and $k$.

cs.LG↗
arXiv2023

Measurement reduction for expectation values via fine-grained commutativity

We introduce a notion of commutativity between operators on a tensor product space, nominally Pauli strings on qubits, that interpolates between qubit-wise commutativity and (full) commutativity. We apply this notion, which we call $k$-commutativity, to measuring expectation values of observables in quantum circuits and show a reduction in the number measurements at the cost of increased circuit depth. Last, we discuss the asymptotic measurement complexity of $k$-commutativity for several families of $n$-qubit Hamiltonians, showing examples with $O(1)$, $O(\sqrt{n})$, and $O(n)$ scaling.

quant-ph↗
arXiv2023

Mathematical conjecture generation using machine intelligence

Conjectures have historically played an important role in the development of pure mathematics. We propose a systematic approach to finding abstract patterns in mathematical data, in order to generate conjectures about mathematical inequalities, using machine intelligence. We focus on strict inequalities of type f < g and associate them with a vector space. By geometerising this space, which we refer to as a conjecture space, we prove that this space is isomorphic to a Banach manifold. We develop a structural understanding of this conjecture space by studying linear automorphisms of this manifold and show that this space admits several free group actions. Based on these insights, we propose an algorithmic pipeline to generate novel conjectures using geometric gradient descent, where the metric is informed by the invariances of the conjecture space. As proof of concept, we give a toy algorithm to generate novel conjectures about the prime counting function and diameters of Cayley graphs of non-abelian simple groups. We also report private communications with colleagues in which some conjectures were proved, and highlight that some conjectures generated using this procedure are still unproven. Finally, we propose a pipeline of mathematical discovery in this space and highlight the importance of domain expertise in this pipeline.

cs.LG↗
arXiv2023

The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems

Qudits with local dimension $d>2$ can have unique structure and uses that qubits ($d=2$) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. We study the structure of the qudit Pauli group for any, including composite, $d$ in several ways. To cover composite values of $d$, we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.

quant-ph↗
arXiv2021

Density theorems with applications in quantum signal processing

We study the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing. Although approximation only in the domain $[0,1]$ is physically desired, these polynomial families are defined by bound constraints not just in $[0,1]$, but also with additional bound constraints outside $[0,1]$. One might wonder then if these additional constraints inhibit their approximation properties within $[0,1]$. The main result of this paper is that this is not the case -- the additional constraints do not hinder the ability of these polynomial families to approximate arbitrarily well any continuous function $f:[0,1] \rightarrow [0,1]$ in the supremum norm, provided $f$ also matches any polynomial in the family at $0$ and $1$. We additionally study the specific problem of approximating the step function on $[0,1]$ (with the step from $0$ to $1$ occurring at $x=\frac{1}{2}$) using one of these families, and propose two subfamilies of monotone and non-monotone approximations. For the non-monotone case, under some additional assumptions, we provide an iterative heuristic algorithm that finds the optimal polynomial approximation.

math.CA↗
arXiv2021

A graph-based formalism for surface codes and twists

Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates. In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the well-defined homological formalism introduced by Kitaev for describing CSS surface codes. In particular, we associate a surface code to any graph $G$ embedded on any 2D-manifold, in such a way that (1) qubits are associated to the vertices of the graph, (2) stabilizers are associated to faces, (3) twist defects are associated to odd-degree vertices. In this way, we are able to reproduce the variety of surface codes, with and without twists, in the literature and produce some new examples. We also calculate and bound various code properties such as the rate and distance in terms of topological graph properties such as genus, systole, and face-width.

quant-ph↗
arXiv2020

The Index of Invariance and its Implications for a Parameterized Least Squares Problem

We study the problem $x_{b,ω} := \text{arg min}_{x \in \mathcal{S}} \|(A + ωI)^{-1/2} (b - Ax)\|_2$, with $A = A^*$, for a subspace $\mathcal{S}$ of $\mathbb{F}^n$ ($\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$), and $ω> -λ_{min}(A)$. We show that there exists a subspace $\mathcal{Y}$ of $\mathbb{F}^n$, independent of $b$, such that $\{x_{b,ω} - x_{b,μ} \mid ω,μ> -λ_{min}(A)\} \subseteq \mathcal{Y}$, where $\dim(\mathcal{Y}) \leq \dim(\mathcal{S} + A\mathcal{S}) - \dim(\mathcal{S}) = \mathbf{Ind}_A(\mathcal{S})$, a quantity which we call the index of invariance of $\mathcal{S}$ with respect to $A$. In particular if $\mathcal{S}$ is a Krylov subspace, this implies the low dimensionality result of Hallman & Gu (2018). The problem is also such that when $A$ is positive and $\mathcal{S}$ is a Krylov subspace, it reduces to CG for $ω= 0$ and to MINRES for $ω\to \infty$. We study several properties of $\mathbf{Ind}_A(\mathcal{S})$ in relation to $A$ and $\mathcal{S}$. We show that the dimension of the affine subspace $\mathcal{X}_b$ containing the solutions $x_{b,ω}$ can be smaller than $\mathbf{Ind}_A(\mathcal{S})$ for all $b$. However, we also exhibit some sufficient conditions on $A$ and $\mathcal{S}$, under which $\mathcal{X} := \text{Span}{\{x_{b,ω} - x_{b,μ} \mid b \in \mathbb{F}^n, ω,μ> -λ_{min}(A)\}}$ has dimension equal to $\mathbf{Ind}_A(\mathcal{S})$. We then study the injectivity of the map $ω\mapsto x_{b,ω}$, leading us to a proof of the convexity result from Hallman & Gu (2018). We finish by showing that sets such as $M(\mathcal{S},\mathcal{S}') = \{A \in \mathbb{F}^{n \times n} \mid \mathcal{S} + A\mathcal{S} = \mathcal{S}'\}$, for nested subspaces $\mathcal{S} \subseteq \mathcal{S}' \subseteq \mathbb{F}^n$, form smooth real manifolds, and explore some topological relationships between them.

math.NA↗
arXiv2019

On sets of commuting and anticommuting Paulis

In this work we study the structure and cardinality of maximal sets of commuting and anticommuting Paulis in the setting of the abelian Pauli group. We provide necessary and sufficient conditions for anticommuting sets to be maximal, and present an efficient algorithm for generating anticommuting sets of maximum size. As a theoretical tool, we introduce commutativity maps, and study properties of maps associated with elements in the cosets with respect to anticommuting minimal generating sets. We also derive expressions for the number of distinct sets of commuting and anticommuting abelian Paulis of a given size.

quant-ph↗