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At least 19 records
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DOE OSTI2026

Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic States

A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for $n$-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of $n$-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the $(n+1)$D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted $\mathbb{Z}_2^3$ gauge theory (equivalent to $\mathbb{D}_4$ topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in $O(d)$ rounds via code switching. In 3D, we construct a transversal logical $\sqrt{\text{T}}$ gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted $\mathbb{Z}_2^4$ gauge theory. Furthermore, our constructions surpass the Bravyi-König bound by achieving the logical gates in the $(n+1)$-th level of Clifford hierarchy in $n$ spatial dimension.

Kobayashi, Ryohei [Institute for Advanced Study, P↗
DOE OSTI2025

Bounding entanglement entropy with Clifford double cosets

Following on our previous work studying the orbits of quantum states under Clifford circuits via reachability graphs, we introduce contracted graphs whose vertices represent classes of quantum states with the same entropy vector. These contracted graphs represent the double cosets of the Clifford group, where the left cosets are built from the stabilizer subgroup of the starting state and the right cosets are built from the entropy-preserving operators. We study contracted graphs for stabilizer states, as well as 𝑊 states and Dicke states, discussing how the diameter of a state's contracted graph constrains the entropic diversity of its two-qubit Clifford orbit. We derive an upper bound on the number of entropy vectors that can be generated using any 𝑛-qubit Clifford circuit, for any quantum state. Here, we speculate on the holographic implications for the relative proximity of gravitational duals of states within the same Clifford orbit. Although we concentrate on how entropy evolves under the Clifford group, our double-coset formalism, and thus the contracted graph picture, is extendable to generic gate sets and generic state properties.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Clifford transformations for fermionic quantum systems: From Pauli and Majorana operators to Dirac fermions

Clifford gates and transformations, which map products of elementary Pauli or Majorana operators to other such products, are foundational in quantum computing, underpinning the stabilizer formalism, error-correcting codes, magic state distillation, quantum communication and cryptography, and qubit tapering. Moreover, circuits composed entirely of Clifford gates are classically simulatable, highlighting their computational significance. In this article we extend the concept of Clifford transformations to Dirac fermions. We demonstrate that discrete Clifford transformations are generated by half-body and pair operators while continuous Clifford transformations are generated by number operators, providing a systematic framework for their characterization. Additionally, we establish connections with fermionic mean-field theories and applications in qubit tapering, offering insights into their broader implications in quantum computing.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2025

Structure of the Majorana Clifford group

In quantum information science, Clifford operators and stabilizer codes play a central role for systems of qubits (or qudits). In this study, we study their analogs for systems composed of Majorana fermions. In this case, a crucial role is played by fermion parity symmetry, which is an unbreakable symmetry present in any system with fundamentally fermionic degrees of freedom. We prove that the subgroup of parity-preserving Majorana Cliffords can be represented by the orthogonal group over the binary field 𝔽 2 , and we show how it can be generated by braiding operators and used to construct any (even-parity) Majorana stabilizer code. We also analyze the frame potential for this so-called p-Clifford group when acting on a fixed-parity sector of the Hilbert space, proving that it is equivalent to the frame potential of the ordinary Clifford group acting on the same sector.

Computational complexity↗
DOE OSTI2025

Synthesis of Single Qutrit Circuits from Clifford + R Gates

The Clifford + R gate-set is a promising basis for fault-tolerant synthesis of qutrit unitaries. We present an algorithm for approximating an arbitrary single-qutrit unitary with a circuit over the Clifford + R gates. Moreover, we analyze its complexity and obtain the non-Clifford gates cost.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Synthesis of single-qutrit circuits from Clifford+𝑅 gates

Here, we present two deterministic compilation algorithms for single-qutrit unitaries with O ( log 1 / ɛ ) gate depth. Each algorithm selects a nearby approximation to the target unitary and then exactly synthesizes the approximation over the Clifford + R basis. The first algorithm exhaustively searches over the group; while the second algorithm searches only for Householder reflections. The exhaustive search algorithm yields an average R count of 2.193 ( 11 ) + 8.621 ( 7 ) log 10 ( 1 / ɛ ) , albeit with a time complexity of O ( ɛ − 4.4 ) . The Householder search algorithm results in a larger average R count of 3.20 ( 13 ) + 10.77 ( 3 ) log 10 ( 1 / ɛ ) at a reduced time complexity of O ( ɛ − 0.42 ) , greatly extending the reach in ɛ . These costs correspond asymptotically to 35% and 69% more non-Clifford gates compared with synthesizing the same unitary with two qubits. Such initial results are encouraging for using the R gate as the nontransversal gate for qutrit-based computation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Any Clifford+T circuit can be controlled with constant T-depth overhead

Since an n-qubit circuit consisting of CNOT gates can have up to $Ω(n^2/\log{n})$ CNOT gates, it is natural to expect that $Ω(n^2/\log{n})$ Toffoli gates are needed to apply a controlled version of such a circuit. We show that the Toffoli count can be reduced to at most n. The Toffoli depth can also be reduced to O(1), at the cost of 2n Toffoli gates, even without using any ancilla or measurement. In fact, using a measurement-based uncomputation, the Toffoli depth can be further reduced to 1. From this, we give two corollaries: any controlled Clifford circuit can be implemented with O(1) T-depth, and any Clifford+T circuit with T-depth D can be controlled with T-depth O(D), even without ancillas. As an application, we show how to catalyze a rotation by any angle up to precision $ε$ in T-depth exactly 1 using a universal $\lceil\log_2(8/ε)\rceil$-qubit catalyst state.

FOS: Physical sciences↗
DOE OSTI2025

Clifford Circuit-Based Heuristic Optimization of Fermion-To-Qubit Mappings

Simulation of interacting Fermionic Hamiltonians is one of the most promising applications of quantum computers. However, the feasibility of analyzing Fermionic systems with a quantum computer hinges on the efficiency of Fermion-to-qubit mappings that encode nonlocal Fermionic degrees of freedom in local qubit degrees of freedom. While recent studies have highlighted the importance of designing Fermion-to-qubit mappings that are tailored to specific problem Hamiltonians, the methods proposed so far either are restricted to a narrow class of mappings or they use computationally expensive and unscalable brute-force search algorithms. Here, in this work, we address this challenge by designing a heuristic numerical optimization framework for Fermion-to-qubit mappings. To this end, we first translate the Fermion-to-qubit mapping problem to a Clifford circuit optimization problem and then use simulated annealing to optimize the average Pauli weight of the problem Hamiltonian. For all Fermionic Hamiltonians we have considered, the numerically optimized mappings outperform their conventional counterparts, including ternary-tree-based mappings that are known to be optimal for single creation and annihilation operators. We find that our optimized mappings yield between 15% and 40% improvements on the average Pauli weight when the simulation Hamiltonian has an intermediate level of complexity. Most remarkably, the optimized mappings improve the average Pauli weight for 6 × 6 nearest-neighbor hopping and Hubbard models by more than 40% and 20%, respectively. Surprisingly, we also find specific interaction Hamiltonians for which the optimized mapping outperforms any ternary-tree-based mapping. Our results establish heuristic numerical optimization as an effective method for obtaining mappings tailored for specific Fermionic Hamiltonian.

Hamiltonians↗
DOE OSTI2024

Realistic Cost to Execute Practical Quantum Circuits using Direct Clifford+T Lattice Surgery Compilation

We report a resource estimation pipeline that explicitly compiles quantum circuits expressed using the Clifford+T gate set into a surface code lattice surgery instruction set. The cadence of magic state requests from the compiled circuit enables the optimization of magic state distillation and storage requirements in a post-hoc analysis. To compile logical circuits into lattice surgery operations, we build upon the open-source Lattice Surgery Compiler. The revised compiler operates in two stages: the first translates logical gates into an abstract, layout-independent instruction set; the second compiles these into local lattice surgery instructions that are allocated to hardware tiles according to a specified resource layout. The second stage retains logical parallelism while avoiding resource contention in the fault-tolerant layer, aiding realism. Additionally, users can specify dedicated tiles at which magic states are replenished, enabling resource costs from the logical computation to be considered independently from magic state distillation and storage. We demonstrate the applicability of our pipeline to large practical quantum circuits by providing resource estimates for the ground state estimation of molecules. Finally, we find that variable magic state consumption rates in real circuits can cause the resource costs of magic state storage to dominate unless production is varied to suit.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Real classical shadows

Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis $\mathcal{W}$ after it has evolved under a unitary transformation randomly sampled from a chosen distribution $\mathcal{U}$. In this work we study the case where $\mathcal{U}$ corresponds to either local or global orthogonal Clifford gates, and $\mathcal{W}$ consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Efficient Simulation of Logical Magic State Preparation Protocols

Developing space- and time-efficient logical magic state preparation (MSP) protocols will likely be an essential step toward building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical MSP protocols under the standard circuit-level noise model. When applied to protocols based on code-switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the nonstabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical MSP protocols without resorting to approximations or resource-intensive state-vector simulations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Evaluating a quantum-classical quantum Monte Carlo algorithm with Matchgate shadows

Solving the electronic structure problem of molecules and solids to high accuracy is a major challenge in quantum chemistry and condensed matter physics. The rapid emergence and development of quantum computers offer a promising route to systematically tackle this problem. Recent work by [Huggins et al ., Nature (London) 603 , 416 (2022)] proposed a hybrid quantum-classical quantum Monte Carlo (QC-QMC) algorithm using Clifford shadows to determine the ground state of a Fermionic Hamiltonian. This approach displayed inherent noise resilience and the potential for improved accuracy compared to its purely classical counterpart. Nevertheless, the use of Clifford shadows introduces an exponentially scaling postprocessing cost. In this work, we investigate an improved QC-QMC scheme utilizing the recently developed Matchgate shadows technique [Commun. Math. Phys. 404 , 629 (2023)], which removes the aforementioned exponential bottleneck. We observe from experiments on quantum hardware that the use of Matchgate shadows in QC-QMC is inherently noise robust. We show that this noise resilience has a more subtle origin than in the case of Clifford shadows. Nevertheless, we find that classical postprocessing, while asymptotically efficient, requires hours of runtime on thousands of classical CPUs for even the smallest chemical systems, presenting a major challenge to the scalability of the algorithm.

Monte Carlo methods↗
DOE OSTI2025

QuCLEAR

SF-25-008 This software optimizes quantum circuits using a two-step process. The first step, Clifford Extraction, moves the Clifford gates to the end of the circuit and includes circuit optimizations. The second step, Clifford Absorption, then addresses these extracted gates on a classical computer. By reducing the number of gates that need to run on the quantum device, the software achieves a significantly lower gate count.

LIU, JI [Argonne National Laboratory (ANL), Argonn↗
DOE OSTI2026

Qudit Designs and Where to Find Them

Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing exponential separations between classical and quantum query complexity. While unitary designs originating from a group structure, such as the Clifford group, have proven to be incredibly useful for qubit systems, unfortunately, this is no longer true for qudits. In fact, the classification of finite-group representations rules out the existence of unitary 2-designs for arbitrary qudit dimensions. This severely limits the applicability of standard quantum information primitives when it comes to qudit systems. We overcome these limitations with a three-fold contribution. First, we introduce a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions. These weighted state-designs generalize classical shadow tomography protocol from qubits to qudits. Second, we introduce a Clifford character RB that allows us to benchmark the qudit Clifford group in any dimension, including non-prime-power dimensions. And third, we establish bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits. Our work further highlights the analogy between spin and optical coherent states by proving that spin-GKP codewords form a state 2-design while spin coherent states do not; in direct analogy with the optical case. This work is structured as a pedagogical and self-contained introduction to unitary designs and their applications to qudit systems.

Anand, Namit [NASA, Ames; Unlisted, US] (ORCID:000↗
DOE OSTI2026

Fault-tolerant resource comparison of qudit and qubit encodings for diagonal quadratic operators

Finite local Hilbert-space truncations arise naturally in quantum simulations of lattice field theories and motivate qudit encodings, but their fault-tolerant advantage over qubit encodings remains unclear. We compare the non-Clifford cost of implementing quadratic diagonal evolutions, exemplified by 𝑈 = 𝑒$^{−𝑖⁢𝑡⁢𝜙^2_𝑥}$ in a uniform field-amplitude discretization of a real scalar field, using either one logical 𝑑-level qudit or 𝑛 𝑏 = ⌈log 2⁡ 𝑑⌉ logical qubits. We analyze two standard settings: product-formula simulation and linear combination of unitaries (LCU) per block encoding, taking the resource metric to be the number of non-Clifford gates after synthesis into a discrete logical gate set. Because tight synthesis bounds for general single-qudit rotations are not known, we express the qudit constructions in terms of embedded two-level SU⁡(2) rotations and derive explicit finite-𝑑 break-even conditions for their synthesis cost; these serve as compiler targets for when qudit encodings can outperform the qubit baseline. Within the constructive models studied here, product-formula implementations would require an exponentially stronger per-primitive synthesis advantage for qudits to win asymptotically, while in the LCU setting the qubit encoding is asymptotically cheaper in 𝑑. Nevertheless, the finite-𝑑 threshold analysis identifies low-dimensional regions in which qudits can yield meaningful constant-factor savings, particularly for LCU-based implementations. As a secondary analysis of the LCU construction, we use an idealized negligible-overhead qubit-qudit code-switching model to give an absolute 𝑇-count comparison and reinterpret the savings as an allowable per-switch overhead budget.

Godwood, Samuel [Univ. of Liverpool (United Kingdo↗