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Matthew Steinberg

Publications and source records attributed to Matthew Steinberg.

17 recordsLinked to original sources
arXiv2026

Quantum codes from classical annealing

We introduce an adaptive simulated annealing algorithm to search for moderately-sized quantum error-correcting codes with high encoding rates and large distances. Our search targets two classes of stabilizer codes: (1) CSS codes, and (2) a subclass of CSS codes that we call ``self-dual with equivalent logicals'' (SWEL) codes, the latter of which which admit transversal implementations of logical Hadamard and phase gates that can be leveraged to construct fault-tolerant gate sets. The search is guided by an energy function that acts as a surrogate for the logical error rate in a code-capacity noise model, combining code distance with a count of minimum-weight logical operators to resolve the discrete plateaux that impede na\"ive distance optimization. For block lengths of up to $50$ physical qubits, our search finds state-of-the-art CSS and SWEL codes whose distances frequently meet or exceed the variants of the quantum Gilbert-Varshamov bound. In addition to providing favorable seed codes for fault-tolerant architectures based on code concatenation, the codes found in this work are promising candidates for high-rate code demonstrations on near-term quantum computing hardware.

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arXiv2026

Synthesis and Optimization of Encoding Circuits for Fault-Tolerant Quantum Computation

Preparing arbitrary logical states is a central primitive for universal fault-tolerant quantum computation and the cost of encoded-state preparation contributes directly to the overall resource overhead. This makes the synthesis of efficient general-state encoding circuits an important problem, particularly with respect to two-qubit gate count and circuit depth. Yet the synthesis of such encoders has been studied less extensively than general Clifford circuit synthesis or the preparation of specific logical Pauli-eigenstates. In this work, we develop methods for synthesizing efficient encoders for arbitrary stabilizer codes. We formulate encoder synthesis as a search over stabilizer tableaus and introduce greedy and rollout-based algorithms that exploit the freedom among stabilizer-equivalent realizations of the same encoding isometry. For code families with a modular structure, such as generalized concatenated and holographic codes, we show how large encoders can be assembled from optimized local constituent encoders, and we use SMT-based exact synthesis to obtain optimal local circuits for small instances. We further evaluate the proposed methods on a broad set of stabilizer codes, including holographic and quantum low-density parity-check (qLDPC) codes, and compare them against recent encoder-synthesis methods and existing constructions from the literature, obtaining improvements of up to 43% in two-qubit gate count and up to 70% in depth. Our results support the optimization of encoded-state preparation in several fault-tolerant quantum-computing schemes, and all methods are openly available as part of the Munich Quantum Toolkit.

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arXiv2026

Digital signatures with classical shadows on near-term quantum computers

Quantum mechanics provides cryptographic primitives whose security is grounded in hardness assumptions independent of those underlying classical cryptography. However, existing proposals require low-noise quantum communication and long-lived quantum memory, capabilities which remain challenging to realize in practice. In this work, we introduce a quantum digital signature scheme that operates with only classical communication, using the classical shadows of states produced by random circuits as public keys. We provide theoretical and numerical evidence supporting the conjectured hardness of learning the private key (the circuit) from the public key (the shadow). A key technical ingredient enabling our scheme is an improved state-certification primitive that achieves higher noise tolerance and lower sample complexity than prior methods. We realize this certification by designing a high-rate error-detecting code tailored to our random-circuit ensemble and experimentally generating shadows for 32-qubit states using circuits with $\geq 80$ logical ($\geq 582$ physical) two-qubit gates, attaining 0.90 $\pm$ 0.01 fidelity. With increased number of measurement samples, our hardware-demonstrated primitives realize a proof-of-principle quantum digital signature, demonstrating the near-term feasibility of our scheme.

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arXiv2025

Certified randomness amplification by dynamically probing remote random quantum states

Cryptography depends on truly unpredictable numbers, but physical sources emit biased or correlated bits. Quantum mechanics enables the amplification of imperfect randomness into nearly perfect randomness, but prior demonstrations have required physically co-located, loophole-free Bell tests, constraining the feasibility of remote operation. Here we realize certified randomness amplification across a network by dynamically probing large, entangled quantum states on Quantinuum's 98-qubit Helios trapped-ion quantum processor. Our protocol is secure even if the remote device acts maliciously or is compromised by an intercepting adversary, provided the samples are generated quickly enough to preclude classical simulation of the quantum circuits. We stream quantum gates in real time to the quantum processor, maintain quantum state coherence for $\approx 0.9$ seconds, and then reveal the measurement bases to the quantum processor only milliseconds before measurement. This limits the time for classical spoofing to 30 ms and constrains the location of hypothetical adversaries to a $4{,}500$ km radius. We achieve a fidelity of 0.586 on random circuits with 64 qubits and 276 two-qubit gates, enabling the amplification of realistic imperfect randomness with a low entropy rate into nearly perfect randomness.

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arXiv2025

Universal fault-tolerant logic with heterogeneous holographic codes

The study of holographic bulk-boundary dualities has led to the construction of novel quantum error correcting codes. Although these codes have shed new light on conceptual aspects of these dualities, they have widely been believed to lack a crucial feature of practical quantum error correction: The ability to support universal fault-tolerant quantum logic. In this work, we introduce a new class of holographic codes that realize this feature. These heterogeneous holographic codes are constructed by combining two seed codes in a tensor network on an alternating hyperbolic tiling. We show how this construction generalizes previous strategies for fault tolerance in tree-type concatenated codes, allowing one to implement non-Clifford gates fault-tolerantly on the holographic boundary. We also demonstrate that these codes allow for high erasure thresholds under a suitable heterogeneous combination of specific seed codes. Compared to previous concatenated codes, heterogeneous holographic codes achieve large overhead savings in physical qubits, e.g., a $21.8\%$ reduction for a two-layer Steane/quantum Reed-Muller combination. Unlike standard concatenated codes, we establish that the new codes can encode more than a single logical qubit per code block by applying ``black hole'' deformations with tunable rate and distance, while possessing fully addressable, universal fault-tolerant gate sets. Therefore, our work strengthens the case for the utility of holographic quantum codes for practical quantum computing.

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arXiv2025

Tensor networks for quantum computing

In the rapidly evolving field of quantum computing, tensor networks serve as an important tool due to their multifaceted utility. In this paper, we review the diverse applications of tensor networks and show that they are an important instrument for quantum computing. Specifically, we summarize the application of tensor networks in various domains of quantum computing, including simulation of quantum computation, quantum circuit synthesis, quantum error correction and mitigation, and quantum machine learning. Finally, we provide an outlook on the opportunities and the challenges of the tensor-network techniques.

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arXiv2024

Near-Term Spin-Qubit Architecture Design via Multipartite Maximally-Entangled States

The design and benchmarking of quantum computer architectures traditionally rely on practical hardware restrictions, such as gate fidelities, control, and cooling. At the theoretical and software levels, numerous approaches have been proposed for benchmarking quantum devices, ranging from, inter alia, quantum volume to randomized benchmarking. In this work, we utilize the quantum information-theoretic properties of multipartite maximally-entangled quantum states, in addition to their correspondence with quantum error correction codes, permitting us to quantify the entanglement generated on near-term bilinear spin-qubit architectures. For this aim, we introduce four metrics which ascertain the quality of genuine multipartite quantum entanglement, along with circuit-level fidelity measures. As part of the task of executing a quantum circuit on a device, we devise simulations which combine expected hardware characteristics of spin-qubit devices with appropriate compilation techniques; we then analyze three different architectural choices of varying lattice sizes for bilinear arrays, under three increasingly realistic noise models. We find that if the use of a compiler is assumed, sparsely-connected spin-qubit lattices can approach comparable values of our metrics to those of the most highly-connected device architecture. Even more surprisingly, by incorporating crosstalk into our last noise model, we find that, as error rates for crosstalk approach realistic values, the benefits of utilizing a bilinear array with advanced connectivity vanish. Our results highlight the limitations of adding local connectivity to near-term spin-qubit devices, and can be readily adapted to other qubit technologies. The framework developed here can be used for analyzing quantum entanglement on a device before fabrication, informing experimentalists on concomitant realistic expectations.

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arXiv2024

LEGO_HQEC: Automating the Analysis, Construction, and Decoding of Holographic Quantum Codes

Quantum error correction (QEC) is a crucial prerequisite for future large-scale quantum computation. Finding and analyzing new QEC codes, along with efficient decoding and fault-tolerance protocols, is central to this effort. Holographic codes are a recent class of generalized concatenated codes derived from holographic bulk/boundary dualities. In addition to exploring the physics of such dualities, these codes possess useful QEC properties such as tunable encoding rates, distance scaling competitive with other well-studied code classes,and excellent recovery thresholds. To allow for a comprehensive analysis of holographic code constructions, we introduce LEGO_HQEC, a software package utilizing the quantum LEGO formalism. This package allows for the construction and analysis of holographic codes on regular hyperbolic tilings, computing their stabilizer generators and logical operators for a specified number of seed codes and layers. Three decoders are included: an erasure decoder based on Gaussian elimination; an integeroptimization decoder; and a tensor-network decoder. With these tools, LEGO_HQEC enables systematic studies of both previously known holographic codes and novel variants. As a demonstration, we provide new numerical results on the holographic blackhole pentagon code, establishing its threshold behavior under the erasure channel as a benchmark example.

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arXiv2024

Quantum Computing and Tensor Networks for Laminate Design: A Novel Approach to Stacking Sequence Retrieval

As with many tasks in engineering, structural design frequently involves navigating complex and computationally expensive problems. A prime example is the weight optimization of laminated composite materials, which to this day remains a formidable task, due to an exponentially large configuration space and non-linear constraints. The rapidly developing field of quantum computation may offer novel approaches for addressing these intricate problems. However, before applying any quantum algorithm to a given problem, it must be translated into a form that is compatible with the underlying operations on a quantum computer. Our work specifically targets stacking sequence retrieval with lamination parameters. To adapt this problem for quantum computational methods, we map the possible stacking sequences onto a quantum state space. We further derive a linear operator, the Hamiltonian, within this state space that encapsulates the loss function inherent to the stacking sequence retrieval problem. Additionally, we demonstrate the incorporation of manufacturing constraints on stacking sequences as penalty terms in the Hamiltonian. This quantum representation is suitable for a variety of classical and quantum algorithms for finding the ground state of a quantum Hamiltonian. For a practical demonstration, we performed state-vector simulations of two variational quantum algorithms and additionally chose a classical tensor network algorithm, the DMRG algorithm, to numerically validate our approach. Although this work primarily concentrates on quantum computation, the application of tensor network algorithms presents a novel quantum-inspired approach for stacking sequence retrieval.

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arXiv2024

Biased-Noise Thresholds of Zero-Rate Holographic Codes with Tensor-Network Decoding

A crucial insight for practical quantum error correction is that different types of errors, such as single-qubit Pauli operators, typically occur with different probabilities. Finding an optimal quantum code under such biased noise is a challenging problem, related to the (generally unknown) maximum capacity of the corresponding noisy channel. A benchmark for this capacity is given by the hashing bound, which describes the performance of random stabilizer codes and leads to the matter of identifying codes that come close to the bound while also being efficiently decodable. In this work, we perform the first comprehensive analysis of asymptotically zero-rate holographic codes under biased noise. We show that many representatives from such models of this code class fulfill both the channel optimality and efficient decoding guarantees for tensor-network codes. In fact, all holographic codes tested were found to reach the hashing bound in some bias regime, while several built from the $\codepar{5,1,2}$ surface code and $\codepar{6,1,3}$ code exceed state-of-the-art code performance in the 2-Pauli noise regime. Furthermore, we consider Clifford deformations which allow all considered codes to reach the hashing bound for 1-Pauli noise as well. Our results establish that holographic codes, which were previously shown to possess efficient tensor-network decoders, also exhibit competitive thresholds under biased noise.

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arXiv2024

Far from Perfect: Quantum Error Correction with (Hyperinvariant) Evenbly Codes

We introduce a new class of qubit codes that we call Evenbly codes, building on a previous proposal of hyperinvariant tensor networks. Its tensor network description consists of local, non-perfect tensors describing CSS codes interspersed with Hadamard gates, placed on a hyperbolic $\{p,q\}$ geometry with even $q\geq 4$, yielding an infinitely large class of subsystem codes. We construct an example for a $\{5,4\}$ manifold and describe strategies of logical gauge fixing that lead to different rates $k/n$ and distances $d$, which we calculate analytically, finding distances which range from $d=2$ to $d \sim n^{2/3}$. Investigating threshold performance under erasure, depolarizing, and pure Pauli noise channels, we find that the code exhibits a depolarizing noise threshold of about 19.1% in the code-capacity model and 50% for pure Pauli and erasure channels under suitable gauges. We also test a constant-rate version with $k/n = 0.125$, finding excellent error resilience (about 40%) under the erasure channel. Recovery rates for these and other settings are studied both under an optimal decoder as well as a more efficient but non-optimal greedy decoder. We also consider generalizations beyond the CSS tensor construction, compute error rates and thresholds for other hyperbolic geometries, and discuss the relationship to holographic bulk/boundary dualities. Our work indicates that Evenbly codes may show promise for practical quantum computing applications.

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arXiv2024

YAQQ: Yet Another Quantum Quantizer -- Design Space Exploration of Quantum Gate Sets using Novelty Search

In the standard circuit model of quantum computation, the number and quality of the quantum gates composing the circuit influence the runtime and fidelity of the computation. The fidelity of the decomposition of quantum algorithms, represented as unitary matrices, to bounded depth quantum circuits depends strongly on the set of gates available for the decomposition routine. To investigate this dependence, we explore the design space of discrete quantum gate sets and present a software tool for comparative analysis of quantum processing units and control protocols based on their native gates. The evaluation is conditioned on a set of unitary transformations representing target use cases on the quantum processors. The cost function considers three key factors: (i) the statistical distribution of the decomposed circuits' depth, (ii) the statistical distribution of process fidelities for the approximate decomposition, and (iii) the relative novelty of a gate set compared to other gate sets in terms of the aforementioned properties. The developed software, YAQQ (Yet Another Quantum Quantizer), enables the discovery of an optimized set of quantum gates through this tunable joint cost function. To identify these gate sets, we use the novelty search algorithm, circuit decomposition techniques, and stochastic optimization to implement YAQQ within the Qiskit quantum simulator environment. YAQQ exploits reachability tradeoffs conceptually derived from quantum algorithmic information theory. Our results demonstrate the pragmatic application of identifying gate sets that are advantageous to popularly used quantum gate sets in representing quantum algorithms. Consequently, we demonstrate pragmatic use cases of YAQQ in comparing transversal logical gate sets in quantum error correction codes, designing optimal quantum instruction sets, and compiling to specific quantum processors.

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arXiv2024

Lightcone Bounds for Quantum Circuit Mapping via Uncomplexity

Efficiently mapping quantum circuits onto hardware is an integral part of the quantum compilation process, wherein a circuit is modified in accordance with the stringent architectural demands of a quantum processor. Many techniques exist for solving the quantum circuit mapping problem, in addition to several theoretical perspectives that relate quantum circuit mapping to problems in classical computer science. This work considers a novel perspective on quantum circuit mapping, in which the routing process of a simplified circuit is viewed as a composition of quantum operations acting on density matrices representing the quantum circuit and processor. Drawing on insight from recent advances in quantum circuit complexity and information geometry, we show that a minimal SWAP-gate count for executing a quantum circuit on a device emerges via the minimization of the distance between quantum states using the quantum Jensen-Shannon divergence, which we dub the lightcone bound. Additionally, we develop a novel initial placement algorithm based on a graph similarity search that selects the partition nearest to a graph isomorphism between interaction and coupling graphs. From these two ingredients, we construct an algorithm for calculating the lightcone bound, which is directly compared alongside the IBM Qiskit compiler for over $600$ realistic benchmark experiments, as well as against a brute-force method for smaller benchmarks. In our simulations, we unambiguously find that neither the brute-force method nor the Qiskit compiler surpasses our bound, signaling utility for estimating minimal overhead when realizing quantum algorithms on constrained quantum hardware. This work also constitutes the first use of quantum circuit uncomplexity to practically-relevant quantum computing. We anticipate that this method may have diverse applicability outside of the scope of quantum information science.

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arXiv2023

Holographic Codes from Hyperinvariant Tensor Networks

Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.

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arXiv2023

Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes

Flag-style fault-tolerance has become a linchpin in the realization of small fault-tolerant quantum-error correction experiments. The flag protocol's utility hinges on low qubit overhead, which is typically much smaller than in other approaches. However, as in most fault-tolerance protocols, the advantages of flag-style error correction come with a tradeoff: fault tolerance can be guaranteed, but such protocols involve high-depth circuits, due to the need for repeated stabilizer measurements. Here, we demonstrate that a dynamic choice of stabilizer measurements, based on past syndromes, and the utilization of elements from the full stabilizer group, leads to flag protocols with lower-depth syndrome-extraction circuits for the [[5,1,3]] code, as well as for the Steane code when compared to the standard methods in flag fault tolerance. We methodically prove that our new protocols yield fault-tolerant lookup tables, and demonstrate them with a pseudothreshold simulation, showcasing large improvements for all protocols when compared to previously-established methods. This work opens the dialogue on exploiting the properties of the full stabilizer group for reducing circuit overhead in fault-tolerant quantum-error correction.

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arXiv2022

Topological-Graph Dependencies and Scaling Properties of a Heuristic Qubit-Assignment Algorithm

The qubit-mapping problem aims to assign and route qubits of a quantum circuit onto a NISQ device in an optimized fashion, with respect to some cost function. Finding an optimal solution to this problem is known to scale exponentially in computational complexity; as such, it is imperative to investigate scalable qubit-mapping solutions for NISQ computation. In this work, a noise-aware heuristic qubit-assignment algorithm (which assigns initial placements for qubits in a quantum algorithm to qubits on a NISQ device, but does not route qubits during the quantum algorithm's execution) is presented and compared against the optimal \textit{brute-force} solution, as well as a trivial qubit assignment, with the aim to quantify the performance of our heuristic qubit-assignment algorithm. We find that for small, connected-graph algorithms, our heuristic-assignment algorithm faithfully lies in between the effective upper and lower bounds given by the brute-force and trivial qubit-assignment algorithms. Additionally, we find that the topological-graph properties of quantum algorithms with over six qubits play an important role in our heuristic qubit-assignment algorithm's performance on NISQ devices. Finally, we investigate the scaling properties of our heuristic algorithm for quantum processors with up to 100 qubits; here, the algorithm was found to be scalable for quantum-algorithms which admit path-like graphs. Our findings show that as the size of the quantum processor in our simulation grows, so do the benefits from utilizing the heuristic qubit-assignment algorithm, under particular constraints for our heuristic algorithm. This work thus characterizes the performance of a heuristic qubit-assignment algorithm with respect to the topological-graph and scaling properties of a quantum algorithm which one may wish to run on a given NISQ device.

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arXiv2022

Conformal Properties of Hyperinvariant Tensor Networks

Hyperinvariant tensor networks (hyMERA) were introduced as a way to combine the successes of perfect tensor networks (HaPPY) and the multiscale entanglement renormalization ansatz (MERA) in simulations of the AdS/CFT correspondence. Although this new class of tensor network shows much potential for simulating conformal field theories arising from hyperbolic bulk manifolds with quasiperiodic boundaries, many issues are unresolved. In this manuscript we analyze the challenges related to optimizing tensors in a hyMERA with respect to some quasiperiodic critical spin chain, and compare with standard approaches in MERA. Additionally, we show two new sets of tensor decompositions which exhibit different properties from the original construction, implying that the multitensor constraints are neither unique, nor difficult to find, and that a generalization of the analytical tensor forms used up until now may exist. Lastly, we perform randomized trials using a descending superoperator with several of the investigated tensor decompositions, and find that the constraints imposed on the spectra of local descending superoperators in hyMERA are compatible with the operator spectra of several minimial model CFTs.

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