arXiv subjects

Amolak Ratan Kalra

Publications and source records attributed to Amolak Ratan Kalra.

11 recordsLinked to original sources

arXiv2026

Stabilizer Ranks, Barnes Wall Lattices and Magic Monotones

In 2024, Kliuchnikov and Schönnenbeck showed a connection between the Barnes Wall lattices, stabilizer states and Clifford operations. In this work, we study their results and relate them to the problem of lower bounding stabilizer ranks. We show the first quantitative lower bound on stabilizer fidelity as a function of stabilizer ranks, which reproduces the linear-by-log lower bound for $χ_δ({|{H}\rangle^{ \otimes n}})$, i.e, on the approximate stabilizer rank of $|H\rangle^{\otimes n}$. In fact, we show that the lower bound holds even when the fidelity between the approximation and ${|H\rangle}^{\otimes n}$ is exponentially small, which is currently the best lower bound in this regime. Next, we define a new magic monotone for pure states, the Barnes Wall norm, and its corresponding approximate variant. We upper bound these monotones by the $CS$-count of state preparation, and also by the stabilizer ranks. In particular, the upper bound given by the $CS$-count is tight, in the sense that we exhibit states that achieve the bound. Apart from these results, we give a Fidelity Amplification algorithm, which provides a trade-off between approximation error and the stabilizer rank. As a corollary, it gives us a way to compose approximate stabilizer decompositions into approximate decompositions of their tensor products. Finally, we provide an alternate, elementary proof of the existence and density of product states with maximal stabilizer ranks, which was first proven by Lovitz and Steffan (2022), where they used results from algebraic geometry.

quant-ph↗

arXiv2026

Buildings for Synthesis with Clifford+R

We study the problem of exact synthesis for the Clifford+R gate set and give the explicit structure of the underlying Bruhat-Tits building for this group. In this process, we also give an alternative proof of the arithmetic nature of this gate set.

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arXiv2026

Enhanced quantum capacity thresholds from symmetry

The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.

quant-ph↗

arXiv2026

Invariant Theory, Magic State Distillation, and Bounds on Classical Codes

We show that the physical consistency of magic state distillation imposes new constraints on the weight enumerators of classical error-correcting codes. We establish that for $|T\rangle$-state distillation protocols based on linear self-orthogonal $GF(4)$ codes, the distillation threshold and noise-suppression exponent are directly determined by the code's simple weight enumerator. By enforcing the physical consistency of the distillation process -- specifically, that the probability of successfully projecting onto the target state must be non-negative -- we derive a new set of constraints on classical weight enumerators. These ``quantum consistency'' constraints prove to be strictly stronger than those derived from classical invariant theory, yielding new upper bounds on the minimum distance of certain classical and quantum codes. Most notably, we show that these new constraints resolve a long-standing open problem in classical coding theory by proving the non-existence of extremal Hermitian self-dual codes over $GF(4)$ with parameters $[12m, 6m, 4m+2]$. Additionally, we use our formalism to perform an exhaustive search of distillation protocols based on linear $GF(4)$ codes with $n < 20$, finding no protocols with thresholds exceeding the 5-qubit code, and we derive analytic upper bounds on the noise-suppression exponents of such distillation routines as a function of $n$.

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arXiv2026

On Error Thresholds for Pauli Channels: Some answers with many more questions

This paper focuses on error thresholds for Pauli channels. We numerically compute lower bounds for the thresholds using the analytic framework of coset weight enumerators pioneered by DiVincenzo, Shor and Smolin in 1998. In particular, we study potential non-additivity of a variety of small stabilizer codes and their concatenations, and report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes. Using insights from this formalism, we estimate the threshold for concatenated repetition codes of large lengths. Finally, for several concatenations of small stabilizer codes we optimize for channels which lead to maximal non-additivity at the hashing point of the corresponding channel. We supplement these results with a discussion on the performance of various stabilizer codes from the perspective of the non-additivity and threshold problem. We report both positive and negative results, and highlight some counterintuitive observations, to support subsequent work on lower bounds for error thresholds.

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arXiv2025

Synthesis and Arithmetic of Single Qutrit Circuits

In this paper we study single qutrit circuits consisting of words over the Clifford$+D$ cyclotomic gate set, where $D=\text{diag}(\pmξ^{a},\pmξ^{b},\pmξ^{c})$, $ξ$ is a primitive $9$-th root of unity and $a,b,c$ are integers. We characterize classes of qutrit unit vectors $z$ with entries in $\mathbb{Z}[ξ, \frac{1}χ]$ based on the possibility of reducing their smallest denominator exponent (sde) with respect to $χ:= 1 - ξ,$ by acting an appropriate gate in Clifford$+D$. We do this by studying the notion of `derivatives mod $3$' of an arbitrary element of $\mathbb{Z}[ξ]$ and using it to study the smallest denominator exponent of $HDz$ where $H$ is the qutrit Hadamard gate and $D$. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford$+D$ gates naturally arise as gates with sde $0$ and $3$ in the group $U(3,\mathbb{Z}[ξ, \frac{1}χ])$ of $3 \times 3$ unitaries with entries in $\mathbb{Z}[ξ, \frac{1}χ]$. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford$+R$ and recover the previous exact synthesis algorithm in \cite{kmm}. The framework developed to formulate qutrit gate synthesis for Clifford$+D$ extends to qudits of arbitrary prime power.

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arXiv2024

Multi-qutrit exact synthesis

We present an exact synthesis algorithm for qutrit unitaries in $\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2πi/3}])$ over the Clifford$+T$ gate set with at most one ancilla. This extends the already known result of qutrit metaplectic gates being a subset of Clifford$+T$ gate set with one ancilla. As an intermediary step, we construct an algorithm to convert 3-level unitaries into multiply-controlled gates, analogous to Gray codes converting 2-level unitaries into multiply-controlled gates. Finally, using catalytic embeddings, we present an algorithm to exactly synthesize unitaries $\mathcal{U}_{3^n}(\mathbb{Z}[1/3,e^{2πi/9}])$ over the Clifford$+T$ gate set with at most 2 ancillas. This, in particular, gives an exact synthesis algorithm of single-qutrit Clifford$+\mathcal{D}$ over the multi-qutrit Clifford$+T$ gate set with at most two ancillas.

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arXiv2023

Normalizing Resistor Networks

Star to mesh transformations are well-known in electrical engineering, and are reminiscent of local complementation for graph states in qudit stabilizer quantum mechanics. This paper describes a rewriting system for resistor circuits over any positive division rig using general star to mesh transformations. We show how these transformations can be organized into a confluent and terminating rewriting system on the category of resistor circuits. Furthermore, based on the recently established connections between quantum and electrical circuits, this paper pushes forward the quest for approachable normal forms for stabilizer quantum circuits.

math.CT↗

arXiv2022

Categories of Kirchhoff relations

It is known that the category of affine Lagrangian relations, AffLagRel_F, over a field, F, of integers modulo a prime p (with p > 2) is isomorphic to the category of stabilizer quantum circuits for p-dits. Furthermore, it is known that electrical circuits (generalized for the field F) occur as a natural subcategory of AffLagRel_F. The purpose of this paper is to provide a characterization of the relations in this subcategory -- and in important subcategories thereof -- in terms of parity-check and generator matrices as used in error detection. In particular, we introduce the subcategory consisting of Kirchhoff relations to be (affinely) those Lagrangian relations that conserve total momentum or equivalently satisfy Kirchhoff's current law. Maps in this subcategory can be generated by electrical components (generalized for the field F): namely resistors, current dividers, and current and voltage sources. The "source" electrical components deliver the affine nature of the maps while current dividers add an interesting quasi-stochastic aspect. We characterize these Kirchhoff relations in terms of parity-check matrices and in addition, characterizes two important subcategories: the deterministic Kirchhoff relations and the lossless relations. The category of deterministic Kirchhoff relations as electrical circuits are generated by resistors circuits. Lossless relations, which are deterministic Kirchhoff, provide exactly the basic hyper-categorical structure of these settings.

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arXiv2020

A Normal Form for Single-Qudit Clifford+$T$ Operators

We propose a normal form for single-qudit gates composed of Clifford and $T$-gates for qudits of odd prime dimension $p\geq 5$. We prove that any single-qudit Clifford+$T$ operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+$T$ operator with minimal $T$-count.

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arXiv2019

Demonstration of the No-Hiding Theorem on the 5 Qubit IBM Quantum Computer in a Category Theoretic Framework

Quantum no-Hiding theorem, first proposed by Braunstein and Pati [Phys. Rev. Lett. 98, 080502 (2007)], was verified experimentally by Samal et al. [Phys. Rev. Lett. 186, 080401 (2011)] using NMR quantum processor. Till then, this fundamental test has not been explored in any of the experimental architecture. Here, we demonstrate the above no-hiding theorem using the IBM 5Q quantum processor. Categorical algebra developed by Coecke and Duncan [New J. Phys. 13, 043016 (2011)] has been used for better visualization of the no-hiding theorem by analyzing the quantum circuit using the ZX calculus. The experimental results confirm the recovery of missing information by the application of local unitary operations on the ancillary qubits.

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