arXiv subjects

Kaushika De Silva

Publications and source records attributed to Kaushika De Silva.

7 recordsLinked to original sources

arXiv2026

From Block-encoding to Generalized Quantum Signal Processing: Principles, Algorithms and Applications

Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.

quant-ph↗

arXiv2026

An Exactness Barrier for ZX-Calculus Optimization of Synthesized Clifford+T Circuits

Gate synthesis and circuit optimization are usually studied separately, and evidence on their interaction is contradictory: ZX-calculus rewriting removes a stable fraction of Solovay-Kitaev circuits, yet almost nothing from number-theoretically synthesized circuits. We show both behaviours follow from a single bound. For any optimizer that preserves the implemented element exactly--including all sound ZX rewriting with extraction--the achievable T-count is bounded below by the denominator exponent of the synthesized ring element. This exactness barrier is computable per instance and separates exact post-processing from approximation-aware resynthesis by a certified factor reaching 101x at recursion depth five. The two behaviours are then the barrier operating at different distances from the floor. For Solovay-Kitaev circuits we prove that the local ZX simplification layer (spider fusion and identity removal) computes exactly the free-product normal form of Z_2 * Z_8, giving exact per-instance compression and, under a calibrated ergodicity hypothesis, a depth-independent limit law confirmed on two independently constructed nets. For number-theoretically synthesized circuits the floor is already saturated: on single-qubit words automated ZX simplification attains it exactly, via a closed-form formula for minimal T-count in terms of phase linkage through the Z-axis normalizer. At two qubits and beyond the same valuation yields unconditional rigidity certificates, which on the quantum-Shannon-decomposition plus gridsynth pipeline certify 99.4-99.9% of the synthesized T-count as incompressible, with rigidity strengthening as accuracy tightens. This explains, and predicts the size of, the near-null optimization recently reported for that pipeline.

quant-ph↗

arXiv2026

Numerical Evaluation of ZX Calculus Optimization for Solovay Kitaev Quantum Circuit Synthesis

Fault-tolerant architectures implement non-Clifford T gates through magic-state distillation, so the T-count of a synthesized circuit dominates its physical cost. The Solovay-Kitaev algorithm approximates any single-qubit unitary from a finite gate set with a sequence length that grows only polylogarithmically in the inverse target error, but it optimizes for numerical convergence rather than circuit economy, and its output carries structural redundancy that a gate-level compiler cannot see. We report a measurement of what diagrammatic post-processing recovers from that redundancy. Twelve hundred random single-qubit targets, spanning the three Pauli rotation families and the general gate U(theta, phi, lambda), are synthesized over Clifford+T at three recursion depths, translated into graph-like ZX-diagrams, simplified by automated rewriting, and extracted back to circuits. Post-processing removes 26.6-30.1% of the total gate count and 18.5-22.2% of the T-count. The absolute saving grows with recursion depth, from about 60 to about 1600 gates, while the fractional saving does not: it rises slightly from the shallowest setting and is then flat across a twenty-five-fold change in circuit length, and by the deepest setting the four target families are no longer distinguishable from one another. Because the rewrite rules preserve the implemented linear map, the approximation error is unchanged. The compile-time cost of the rewriting layer, by contrast, grows sharply with depth and comes to dominate the synthesis itself.

quant-ph↗

arXiv2026

Hermitian Matrix Function Synthesis without Block-Encoding

Implementing polynomial functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advanced quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding and avoids the compounding post-selection overheads characteristic of LCU-based constructions, achieving a stable, degree-independent success probability. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead and opens new pathways for quantum algorithm design for functions of Hermitian matrices, particularly in settings where the Hermitian operator arises naturally from symmetric combinations of unitaries.

quant-ph↗

arXiv2025

Stochastic quantum adiabatic algorithm with fractional Brownian motion

Adiabatic Quantum Computing relies on the quantum adiabatic theorem, which states that a quantum system evolves along its ground state with time if the governing Hamiltonian varies infinitely slowly. However, practical limitations force computations to be performed within limited times, exposing the system to transitions into excited states, and thereby reducing the success probability. Here we investigate the counterintuitive hypothesis that incorporating stochastic noise, specifically noise driven by fractional Brownian motion, in a non-Markovian setup can enhance the performance of adiabatic quantum computing by improving its success probability at limited evolution times. The study begins by developing the mathematical framework to introduce stochastic noise multiplicatively into the Schrödinger equation, resulting in a stochastic Schrödinger equation. To preserve Itô integrability within the non-Markovian framework, a semimartingale approximation for fractional Brownian motion is employed. We perform numerical simulations to compare the performance of the quantum adiabatic algorithm with and without noise driven by fractional Brownian motion using the NP-complete Exact Cover-3 problem, transformed into the Ising model. Our results exhibit an improvement in success probability in the presence of noise driven by fractional Brownian motion with Hurst parameter $0<H<\frac{1}{2}$ and an increase in speedup as $H$ approaches 0. Although simulations are limited to problems involving a modest number of qubits, evidence suggests that the proposed approach scales favorably with the system size.

quant-ph↗

arXiv2016

A note on two Conjectures on Dimension funcitons of $C^{*}$-algebras

\noindent Let $A$ be an arbitrary $C^*$ algebra. In \cite{BH} Blackadar and Handelman conjectured the set of lower semicontinuous dimension functions on $A$ to be pointwise dense in the set $DF(A)$ of all dimension functions on $A$ and $DF(A)$ to be a Choquet simplex. We provide an equivalent condition for the first conjecture for unital $A$. Then by applying this condition we confirm the first Conjecture for all unital $A$ for which either the radius of comparison is finite or the semigroup $W(A)$ is almost unperforated. As far as we know the most general results on the first Conjecture up to now assumes exactness, simplicity and moreover stronger regularity properties such as strict comparison. Our results are achieved through applications of the techniques developed in \cite{BR} and \cite{R}. We also note that, whenever the first Conjecture holds for some unital $A$ and extreme boundary of the the quasitrace simplex of $A$ is finite, then every dimension function of $A$ is lower semicontinuous and $DF(A)$ is affinely homeomorphic to the quasitrace simplex of $A$. Combing this with the said results on the first Conjecture give us a class of algebras for which $DF(A)$ is a Choquet simplex, i.e. gives a new class for which the 2nd Conjecture mentioned above holds.

math.OA↗

arXiv2015

Rank Constrained Homotopies

For any $n\geq k\geq l\in\mathbb{N},$ let $S(n,k,l)$ be the set of all those non-negative definite matrices $a\in M_{n}(\mathbb{C})$ with $l\leq\text{rank }a\leq k$. Motivated by applications to $C^{*}$-algebra theory, we investigate the homotopy properties of continuous maps from a compact Hausdorff space $X$ into sets of the form $S(n,k,l).$ It is known that for any $n,$ if $k-l$ is approximately 4 times the covering dimension of $X$ then there is only one homotopy class of maps from $X$ into $S(n,k,l)$, i.e. $C(X,S(n,k,l))$ is path connected. In our main Theorem we improve this bound by a factor of 8. By combining classical homotopy theory methods with $C^{*}$-algebraic techniques we also show that if $π_{r}(S(n,k,l))$ vanishes for all $r\leq d$ then $C(X,S(n,k,l))$ is path connected for any compact Hausdorff $X$ with covering dimension not greater than $d$.

math.OA↗