arXiv subjects

Narayanan Rengaswamy

Publications and source records attributed to Narayanan Rengaswamy.

34 records · Page 2Linked to original sources

arXiv2021

Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers

Physical platforms such as trapped ions suffer from coherent noise where errors manifest as rotations about a particular axis and can accumulate over time. We investigate passive mitigation through decoherence free subspaces, requiring the noise to preserve the code space of a stabilizer code, and to act as the logical identity operator on the protected information. Thus, we develop necessary and sufficient conditions for all transversal $Z$-rotations to preserve the code space of a stabilizer code, which require the weight-$2$ $Z$-stabilizers to cover all the qubits that are in the support of some $X$-component. Further, the weight-$2$ $Z$-stabilizers generate a direct product of single-parity-check codes with even block length. By adjusting the size of these components, we are able to construct a large family of QECC codes, oblivious to coherent noise, that includes the $[[4L^2, 1, 2L]]$ Shor codes. Moreover, given $M$ even and any $[[n,k,d]]$ stabilizer code, we can construct an $[[Mn, k, \ge d]]$ stabilizer code that is oblivious to coherent noise. If we require that transversal $Z$-rotations preserve the code space only up to some finite level $l$ in the Clifford hierarchy, then we can construct higher level gates necessary for universal quantum computation. The $Z$-stabilizers supported on each non-zero $X$-component form a classical binary code C, which is required to contain a self-dual code, and the classical Gleason's theorem constrains its weight enumerator. The conditions for a stabilizer code being preserved by transversal $2π/2^l$ $Z$-rotations at $4 \le l \le l_{\max} <\infty$ level in the Clifford hierarchy lead to generalizations of Gleason's theorem that may be of independent interest to classical coding theorists.

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arXiv2021

Classical Coding Problem from Transversal $T$ Gates

Universal quantum computation requires the implementation of a logical non-Clifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical $T$ and $T^{-1}$ gates. For example, this could enable magic state distillation with non-CSS codes and, thus, provide better parameters than CSS-based protocols. However, among non-degenerate stabilizer codes that support transversal $T$, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal $T$ via physical transversal $T$. Using our algebraic approach, we reveal new purely-classical coding problems that are intimately related to the realization of logical operations via transversal $T$. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum Reed-Muller codes. This result is generalized to finer angle $Z$-rotations in arXiv:1910.09333.

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arXiv2021

On Optimality of CSS Codes for Transversal $T$

In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal $T$ and $T^{-1}$ gates preserve the codespace. Our Heisenberg perspective reduces this to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries: (a) For any non-degenerate $[[ n,k,d ]]$ stabilizer code supporting a physical transversal $T$, there exists an $[[ n,k,d ]]$ CSS code with the same property; (b) Triorthogonal codes are the most general CSS codes that realize logical transversal $T$ via physical transversal $T$; (c) Triorthogonality is necessary for physical transversal $T$ on a CSS code to realize the logical identity. The main tool we use is a recent efficient characterization of certain diagonal gates in the Clifford hierarchy (arXiv:1902.04022). We refer to these gates as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing code constructions that realize logical gates via transversal $T$. We provide several examples and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal $π/2^{\ell}$ $Z$-rotations. In particular, using Ax's theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal $π/2^{\ell}$ $Z$-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal $1$- or $2$-local diagonal gate.

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arXiv2021

Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits.

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arXiv2021

Logical Clifford Synthesis for Stabilizer Codes

Quantum error-correcting codes are used to protect qubits involved in quantum computation. This process requires logical operators, acting on protected qubits, to be translated into physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in $\mathbb{C}^{N \times N}$ as a partial $2m \times 2m$ binary symplectic matrix, where $N = 2^m$. We state and prove two theorems that use symplectic transvections to efficiently enumerate all binary symplectic matrices that satisfy a system of linear equations. As a corollary of these results, we prove that for an $[\![ m,k ]\!]$ stabilizer code every logical Clifford operator has $2^{r(r+1)/2}$ symplectic solutions, where $r = m-k$, up to stabilizer degeneracy. The desired physical circuits are then obtained by decomposing each solution into a product of elementary symplectic matrices, that correspond to elementary circuits. This enumeration of all physical realizations enables optimization over the ensemble with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer of the code can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. Our method of circuit synthesis can be applied to any stabilizer code, and this paper discusses a proof of concept synthesis for the $[\![ 6,4,2 ]\!]$ CSS code. Programs implementing the algorithms in this paper, which includes routines to solve for binary symplectic solutions of general linear systems and our overall LCS (logical circuit synthesis) algorithm, can be found at: https://github.com/nrenga/symplectic-arxiv18a

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arXiv2021

Approximate Unitary 3-Designs from Transvection Markov Chains

Unitary $k$-designs are probabilistic ensembles of unitary matrices whose first $k$ statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical $\mathbb{Z}_4$-linear Kerdock codes maps to a unitary $2$-design, which established a new classical-quantum connection via graph states. In this paper, we construct a Markov process that mixes this Kerdock $2$-design with symplectic transvections, and show that this process produces an $ε$-approximate unitary $3$-design. We construct a graph whose vertices are Pauli matrices, and two vertices are connected by directed edges if and only if they commute. A unitary ensemble that is transitive on vertices, edges, and non-edges of this Pauli graph is an exact $3$-design, and the stationary distribution of our process possesses this property. With respect to the symmetries of Kerdock codes, the Pauli graph has two types of edges; the Kerdock $2$-design mixes edges of the same type, and the transvections mix the types. More precisely, on $m$ qubits, the process samples $O(\log(N^5/ε))$ random transvections, where $N = 2^m$, followed by a random Kerdock $2$-design element and a random Pauli matrix. Hence, the simplicity of the protocol might make it attractive for several applications. From a hardware perspective, $2$-qubit transvections exactly map to the Mølmer-Sørensen gates that form the native $2$-qubit operations for trapped-ion quantum computers. Thus, it might be possible to extend our work to construct an approximate $3$-design that only involves such $2$-qubit transvections.

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arXiv2021

Belief Propagation with Quantum Messages for Quantum-Enhanced Classical Communications

For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity achievable with a receiver that performs individual pulse-by-pulse detection, and the quantum-optimal "joint-detection receiver" that acts collectively on long codeword-blocks of modulated pulses; an effect often termed "superadditive capacity". In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of "belief-propagation with quantum messages" (BPQM) (arXiv:1607.04833). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best "classical" approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. Our result suggests that a BPQM-receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel. Moreover, our receiver circuit provides an alternative proposal for a quantum supremacy experiment, targeted at a specific application that can potentially be implemented on a small, special-purpose, photonic quantum computer capable of performing cat-basis universal qubit logic.

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arXiv2021

A Semiclassical Proof of Duality Between the Classical BSC and the Quantum PSC

In 2018, Renes [IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 577-592 (2018)] (arXiv:1701.05583) developed a general theory of channel duality for classical-input quantum-output (CQ) channels. That result showed that a number of well-known duality results for linear codes on the binary erasure channel could be extended to general classical channels at the expense of using dual problems which are intrinsically quantum mechanical. One special case of this duality is a connection between coding for error correction (resp. wire-tap secrecy) on the quantum pure-state channel (PSC) and coding for wire-tap secrecy (resp. error correction) on the classical binary symmetric channel (BSC). While this result has important implications for classical coding, the machinery behind the general duality result is rather challenging for researchers without a strong background in quantum information theory. In this work, we leverage prior results for linear codes on PSCs to give an alternate derivation of the aforementioned special case by computing closed-form expressions for the performance metrics. The noted prior results include optimality of the square-root measurement (SRM) for linear codes on the PSC and the Fourier duality of linear codes. We also show that the SRM forms a suboptimal measurement for channel coding on the BSC (when interpreted as a CQ problem) and secret communications on the PSC. Our proofs only require linear algebra and basic group theory, though we use the quantum Dirac notation for convenience.

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arXiv2020

Un-Weyl-ing the Clifford Hierarchy

The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.

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arXiv2020

On Cyclic Polar Codes and The Burst Erasure Performance of Spatially-Coupled LDPC Codes

Polar codes were introduced in 2009 and proven to achieve the symmetric capacity of any binary-input discrete memoryless channel under low-complexity successive cancellation decoding. In this thesis, we construct cyclic polar codes based on a mixed-radix Cooley-Tukey decomposition of the Galois field Fourier transform. The main results are: we can, for the first time, construct, encode and decode polar codes that are cyclic, with their blocklength being arbitrary; for a given target block erasure rate, we can achieve significantly higher code rates on the erasure channel than the original polar codes, at comparable blocklengths; on the symmetric channel with only errors, we can perform much better than equivalent rate Reed-Solomon codes with the same blocklength, by using soft-decision decoding; and, since the codes are subcodes of higher rate RS codes, a RS decoder can be used if suboptimal performance suffices for the application as a trade-off for higher decoding speed. In 2010, it was shown that spatially-coupled low-density parity-check (LDPC) codes approach the capacity of binary memoryless channels, asymptotically, with belief-propagation (BP) decoding. In this thesis, we are interested in the finite length average performance of randomly coupled LDPC ensembles on binary erasure channels with memory. The significant contributions of this work are: tight lower bounds for the block erasure probability ($P_B$) under various scenarios for the burst pattern; bounds focused on practical scenarios where a burst affects exactly one of the coupled codes; expected error floor for the bit erasure probability ($P_b$) on the binary erasure channel; and, characterization of the performance of random regular ensembles, on erasure channels, with a single vector describing distinct types of size-$2$ stopping sets. All these results are verified using Monte-Carlo simulations.

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arXiv2020

Classical Coding Approaches to Quantum Applications

Quantum information science strives to leverage the quantum-mechanical nature of our universe in order to achieve large improvements in certain information processing tasks. In deep-space optical communications, current receivers for the pure-state classical-quantum channel first measure each qubit channel output and then classically post-process the measurements. This approach is sub-optimal. In this dissertation we investigate a recently proposed quantum algorithm for this task, which is inspired by classical belief-propagation algorithms, and analyze its performance on a simple $5$-bit code. We show that the algorithm is optimal for each bit and it appears to achieve optimal performance when deciding the full transmitted message. We also provide explicit circuits for the algorithm in terms of standard gates. This suggests a near-term quantum communication advantage over the aforementioned sub-optimal scheme. Quantum error correction is vital to building a universal fault-tolerant quantum computer. We propose an efficient algorithm that can translate a given logical Clifford operation on a stabilizer code into all (equivalence classes of) physical Clifford circuits that realize that operation. In order to achieve universality, one also needs to implement at least one non-Clifford logical operation. So, we develop a mathematical framework for a large subset of diagonal operations in the Clifford hierarchy, which we call Quadratic Form Diagonal (QFD) gates. Then we use the QFD formalism to characterize all stabilizer codes whose code spaces are preserved under the transversal action of the non-Clifford $T$ gates on the physical qubits. We also discuss a few purely-classical coding problems motivated by transversal $T$ gates. A conscious effort has been made to keep this dissertation self-contained, by including necessary background material on quantum information and computation.

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arXiv2019

Adaptive Procedures for Discrimination Between Arbitrary Tensor-Product Quantum States

Discrimination between quantum states is a fundamental task in quantum information theory. Given two arbitrary tensor-product quantum states (TPQS) $ρ_{\pm} = ρ_{\pm}^{(1)} \otimes \cdots \otimes ρ_{\pm}^{(N)}$, determining the joint $N$-system measurement to optimally distinguish between the two states is a hard problem. Thus, there is great interest in identifying local measurement schemes that are optimal or close-to-optimal. In this work, we focus on distinguishing between two general TPQS. We begin by generalizing previous work by Acin et al. (Phys. Rev. A 71, 032338) to show that a locally greedy (LG) scheme using Bayesian updating can optimally distinguish between two states that can be written as tensor products of arbitrary pure states. Then, we show that even in the limit of large $N$ the same algorithm cannot distinguish tensor products of mixed states with vanishing error probability. This poor asymptotic behavior occurs because the Helstrom measurement becomes trivial for sufficiently biased priors. Based on this, we introduce a modified locally greedy (MLG) scheme with strictly better performance. In the second part of this work, we compare these simple local schemes with a general dynamic programming (DP) approach that finds the optimal series of local measurements to distinguish the two states. When the subsystems are non-identical, we demonstrate that the ordering of the systems affects performance and we extend the DP technique to determine the optimal ordering adaptively. Finally, in contrast to the binary optimal collective measurement, we show that adaptive protocols on sufficiently large (e.g., qutrit) subsystems must contain non-binary measurements to be optimal. (The code that produced the simulation results in this paper can be found at: https://github.com/SarahBrandsen/AdaptiveStateDiscrimination)

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arXiv2019

Unifying the Clifford Hierarchy via Symmetric Matrices over Rings

The Clifford hierarchy is a foundational concept for universal quantum computation (UQC). It was introduced to show that UQC can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. (arXiv:1608.06596) recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal gates whose action on a computational basis qudit state is described by a $2^k$-th root of unity raised to a polynomial function of the state, and they established the level of such unitaries in the hierarchy. For qubit systems, we consider $k$-th level diagonal gates that can be described just by quadratic forms of the state over the ring $\mathbb{Z}_{2^k}$ of integers mod $2^k$. These involve symmetric matrices over $\mathbb{Z}_{2^k}$ that can be used to efficiently describe all $2$-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal gates from lower levels. This involves symplectic matrices over $\mathbb{Z}_{2^k}$ and hence our perspective unifies these gates with the binary symplectic framework for Clifford gates. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+$T$ set of gates, perhaps leading to shorter depth circuits. Our results suggest that some non-diagonal gates might be understood by generalizing other binary symplectic matrices to integer rings.

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arXiv2018

Synthesis of Logical Clifford Operators via Symplectic Geometry

Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in $\mathbb{C}^{N \times N}$ as a partial $2m \times 2m$ binary symplectic matrix, where $N = 2^m$. We state and prove two theorems that use symplectic transvections to efficiently enumerate all symplectic matrices that satisfy a system of linear equations. As an important corollary of these results, we prove that for an $[\![ m,m-k ]\!]$ stabilizer code every logical Clifford operator has $2^{k(k+1)/2}$ symplectic solutions. The desired physical circuits are then obtained by decomposing each solution as a product of elementary symplectic matrices. Our assembly of the possible physical realizations enables optimization over them with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer of the code can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the $[\![ 6,4,2 ]\!]$ CSS code. We conclude with a classical coding-theoretic perspective for constructing logical Pauli operators for CSS codes. Since our circuit synthesis algorithm builds on the logical Pauli operators for the code, this paper provides a complete framework for constructing all logical Clifford operators for CSS codes. Programs implementing our algorithms can be found at https://github.com/nrenga/symplectic-arxiv18a

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arXiv2018

Finite-Length Analysis of Spatially-Coupled Regular LDPC Ensembles on Burst-Erasure Channels

Regular spatially-Coupled LDPC (SC-LDPC) ensembles have gained significant interest since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this work, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per codeword and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provide new tight lower bounds for the block erasure probability ($P_{\mathrm{B}}$) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. We further show that expurgating the ensemble can improve the block erasure probability by several orders of magnitude. Later we extend our methodology to more general channel models. In a first extension, we consider bursts that can start at a random position in the codeword and span across multiple spatial positions. Besides the finite length analysis, we determine by means of density evolution the maximnum correctable burst length. In a second extension, we consider the case where in addition to a single burst, random bit erasures may occur. Finally, we consider a block erasure channel model which erases each spatial position independently with some probability $p$, potentially introducing multiple bursts simultaneously. All results are verified using Monte-Carlo simulations.

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arXiv2016

Spatially Coupled LDPC Codes Affected by a Single Random Burst of Erasures

Spatially-Coupled LDPC (SC-LDPC) ensembles achieve the capacity of binary memoryless channels (BMS), asymptotically, under belief-propagation (BP) decoding. In this paper, we study the BP decoding of these code ensembles over a BMS channel and in the presence of a single random burst of erasures. We show that in the limit of code length, codewords can be recovered successfully if the length of the burst is smaller than some maximum recoverable burst length. We observe that the maximum recoverable burst length is practically the same if the transmission takes place over binary erasure channel or over binary additive white Gaussian channel with the same capacity. Analyzing the stopping sets, we also estimate the decoding failure probability (the error floor) when the code length is finite.

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