arXiv subjects

Isaac H. Kim

Publications and source records attributed to Isaac H. Kim.

At least 19 recordsLinked to original sources

arXiv2026

Extensive long-range magic in non-Abelian topological orders

We show that the low-energy states of non-Abelian topological orders possess extensive magic which is long-ranged, and cannot be eliminated by a constant-depth local unitary circuit. This refines conventional notions of complexity beyond the linear circuit depth which is required to prepare any topological phase, and provides a new resource-theoretic characterization of topological orders. A central technical result is a no-go theorem establishing that stabilizer states--even up to constant-depth local unitarie--cannot approximate low-energy states of non-Abelian string-net models which satisfy the entanglement bootstrap axioms. Moreover, we show that stabilizer-realizable Abelian string-net phases have mutual braiding phases quantized by the on-site qudit dimension, and that any violation of this condition necessarily implies extensive long-range magic. Extending to higher spatial dimensions, we argue that any state obeying an entanglement area law and hosting excitations with nontrivial fusion spaces must exhibit extensive long-range magic. This applies, in particular, to ground-states and low-energy states of higher-dimensional quantum double models.

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arXiv2026

Sdim: A Qudit Stabilizer Simulator

Quantum computers have steadily improved over the last decade, but developing fault-tolerant quantum computing (FTQC) techniques, required for useful, universal computation remains an ongoing effort. Key elements of FTQC such as error-correcting codes and decoding are supported by a rich bed of stabilizer simulation software such as Stim and CHP, which are essential for numerically characterizing these protocols at realistic scales. Recently, experimental groups have built nascent high-dimensional quantum hardware, known as qudits, which have a myriad of attractive properties for algorithms and FTQC. Despite this, there are no widely available qudit stabilizer simulators. We introduce the first open-source realization of such a simulator for all dimensions. We demonstrate its correctness against existing state vector simulations and benchmark its performance in evaluating and sampling quantum circuits. This simulator is the essential computational infrastructure to explore novel qudit error correction as earlier stabilizer simulators have been for qubits.

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arXiv2025

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.

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arXiv2025

Classical Estimation of the Free Energy and Quantum Gibbs Sampling from the Markov Entropy Decomposition

We revisit the Markov Entropy Decomposition, a classical convex relaxation algorithm introduced by Poulin and Hastings to approximate the free energy in quantum spin lattices. We identify a sufficient condition for its convergence, namely the decay of the effective interaction. The effective interaction, also known as Hamiltonians of mean force, is a widely established correlation measure, and we show our decay condition in 1D at any temperature as well as in the high-temperature regime under a certain commutativity condition on the Hamiltonian building on existing results. This yields polynomial and quasi-polynomial time approximation algorithms in these settings, respectively. Furthermore, the decay of the effective interaction implies the decay of the conditional mutual information for the Gibbs state of the system. We then use this fact to devise a rounding scheme that maps the solution of the convex relaxation to a global state and show that the scheme can be efficiently implemented on a quantum computer, thus proving efficiency of quantum Gibbs sampling under our assumption of decay of the effective interaction.

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arXiv2025

Efficient simulation of logical magic state preparation protocols

Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation 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 non-stabilizerness, 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 magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.

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arXiv2025

Experimental Demonstration of High-Fidelity Logical Magic States from Code Switching

Preparation of high-fidelity logical magic states has remained as a necessary but daunting step towards building a large-scale fault-tolerant quantum computer. One approach is to fault-tolerantly prepare a magic state in one code and then switch to another, a method known as code switching. We experimentally demonstrate this protocol on an ion-trap quantum processor, yielding a logical magic state encoded in an error-correcting code with state-of-the-art logical fidelity. Our experiment is based on the first demonstration of code switching between color codes, from the fifteen-qubit quantum Reed-Muller code to the seven-qubit Steane code. We prepare an encoded magic state in the Steane code with $82.58\%$ probability, with an infidelity of at most $5.1(2.7) \times 10^{-4}$. The reported infidelity is lower than the leading infidelity of the physical operations utilized in the protocol by a factor of at least $2.7$, indicating the quantum processor is below the pseudo-threshold. Furthermore, we create two copies of the magic state in the same quantum processor and perform a logical Bell basis measurement for a sample-efficient certification of the encoded magic state. The high-fidelity magic state can be combined with the already-demonstrated fault-tolerant Clifford gates, state preparation, and measurement of the 2D color code, completing a universal set of fault-tolerant computational primitives with logical error rates equal or better than the physical two-qubit error rate.

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arXiv2025

A quantum algorithm for modular flow

Entanglement is a defining property of quantum systems. For a subsystem of a larger quantum system, one can formally define an operator known as the modular Hamiltonian, which is closely linked to the entanglement properties of that subsystem, and a corresponding operator flow called the modular flow. Algorithms for estimating the von Neumann entropy, the best-known entanglement measure, are well-established, but no equivalent procedures have been previously described for the modular flow. In this work, we briefly review the quantum singular value transform (QSVT) framework for developing quantum algorithms, and then discuss the implementation of modular flow within this framework. We conclude by describing select applications of our modular flow algorithm, such as extracting the chiral central charge of a topologically ordered system and simulating the experience of the bulk observer in holography. We also prove a query complexity lower bound for modular flow, which shows that our method cannot be improved further substantially.

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arXiv2025

Catalytic $z$-rotations in constant $T$-depth

We show that the $T$-depth of any single-qubit $z$-rotation can be reduced to $3$ if a certain catalyst state is available. To achieve an $\epsilon$-approximation, it suffices to have a catalyst state of size polynomial in $\log(1/\epsilon)$. This implies that $\mathsf{QNC}^0_f/\mathsf{qpoly}$ admits a finite universal gate set consisting of Clifford+$T$. In particular, there are catalytic constant $T$-depth circuits that approximate multi-qubit Toffoli, adder, and quantum Fourier transform arbitrarily well. We also show that the catalyst state can be prepared in time polynomial in $\log (1/\epsilon)$.

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arXiv2025

Code switching revisited: Low-overhead magic state preparation using color codes

We propose a protocol to prepare a high-fidelity magic state on a two-dimensional (2D) color code using a three-dimensional (3D) color code. Our method modifies the known code switching protocol with (i) a recently discovered transversal gate between the 2D and the 3D code and (ii) a judicious use of flag-based postselection. We numerically demonstrate that these modifications lead to a significant improvement in the fidelity of the magic state. For instance, subjected to a uniform circuit-level noise of $10^{-3}$ (excluding idling noise), our code switching protocol yields a magic state encoded in the distance-$3$ 2D color code with a logical infidelity of $4.6\times 10^{-5}\pm 1.6 \times 10^{-5}$ (quantified by an error-corrected logical state tomography) with an $84\%$ of acceptance rate. Used in conjunction with a postselection approach, extrapolation from a polynomial fit suggests a fidelity improvement to $5.1 \times 10^{-7}$ for the same code. Our protocol is aimed for architectures that allow nonlocal connectivity and should be readily implementable in near-term devices. Finally, we also present a simulation technique akin to an extended stabilizer simulator which effectively incorporates the non-Clifford $T$-gate, that permits to simulate the protocol without resorting to a resource intensive state-vector simulation.

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arXiv2025

Fault-tolerant Quantum Error Correction Using a Linear Array of Emitters

We propose a fault-tolerant quantum error correction architecture consisting of a linear array of emitters and delay lines. In our scheme, a resource state for fault-tolerant quantum computation is generated by letting the emitters interact with a stream of photons and their neighboring emitters. Depending on the number of emitters $n_e$, we study the effect of delay line errors in two regimes: when $n_e$ is a small constant of order unity and when $n_e$ scales with the code distance. Between these two regimes, the logical error rate steadily decreases as $n_e$ increases, from a scaling of $\exp(-cη^{-1/2})$ to $\exp(-c'η^{-1})$, where $η$ is the error rate per unit length in the delay line, for some constants $c,c'>0$. We also carry out a detailed study of the break-even point and the fault-tolerance overhead. These studies suggest that the multi-emitter architecture, using the state-of-the-art delay lines, can be used to demonstrate error suppression, assuming other sources of errors are sufficiently small.

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arXiv2025

Chirality, magic, and quantum correlations in multipartite quantum states

We introduce a notion of chirality for generic quantum states. A chiral state is defined as a state which cannot be transformed into its complex conjugate in a local product basis using local unitary operations. We introduce a number of quantitative measures of chirality which vanish for non-chiral states. A faithful measure called the "chiral log-distance" is defined in terms of the maximum fidelity between the state and its complex conjugate under local unitary transformations. We further introduce a set of computable measures that involve nested commutators of modular Hamiltonians. We show general relations between chirality and the amount of "magic" or non-stabilizer resource in a state. We first show that stabilizer states of qubits are non-chiral. Furthermore, magic monotones such as the stabilizer nullity and the stabilizer fidelity are lower-bounded by the chiral log-distance. We further relate chirality to discord-like quantum correlations. We define a measure of discord-like correlations called the intrinsic interferometric power, and show that it is lower-bounded by a nested commutator chirality measure. One interesting aspect of chirality is that it is not monotonic under local partial traces, indicating that it is a unique kind of resource that is not captured by traditional quantum resource theories.

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arXiv2025

Conformal geometry from entanglement

In a physical system with conformal symmetry, observables depend on cross-ratios, measures of distance invariant under global conformal transformations (conformal geometry for short). We identify a quantum information-theoretic mechanism by which the conformal geometry emerges at the gapless edge of a 2+1D quantum many-body system with a bulk energy gap. We introduce a novel pair of information-theoretic quantities $(\mathfrak{c}_{\mathrm{tot}}, η)$ that can be defined locally on the edge from the wavefunction of the many-body system, without prior knowledge of any distance measure. We posit that, for a topological groundstate, the quantity $\mathfrak{c}_{\mathrm{tot}}$ is stationary under arbitrary variations of the quantum state, and study the logical consequences. We show that stationarity, modulo an entanglement-based assumption about the bulk, implies (i) $\mathfrak{c}_{\mathrm{tot}}$ is a non-negative constant that can be interpreted as the total central charge of the edge theory. (ii) $η$ is a cross-ratio, obeying the full set of mathematical consistency rules, which further indicates the existence of a distance measure of the edge with global conformal invariance. Thus, the conformal geometry emerges from a simple assumption on groundstate entanglement. We show that stationarity of $\mathfrak{c}_{\mathrm{tot}}$ is equivalent to a vector fixed-point equation involving $η$, making our assumption locally checkable. We also derive similar results for 1+1D systems under a suitable set of assumptions.

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arXiv2025

Chiral Virasoro algebra from a single wavefunction

Chiral edges of 2+1D systems can have very robust emergent conformal symmetry. When the edge is purely chiral, the Hilbert space of low-energy edge excitations can form a representation of a single Virasoro algebra. We propose a method to systematically extract the generators of the Virasoro algebra from a single ground state wavefunction, using entanglement bootstrap and an input from the edge conformal field theory. We corroborate our construction by numerically verifying the commutation relations of the generators. We also study the unitary flows generated by these operators, whose properties (such as energy and state overlap) are shown numerically to agree with our analytical predictions.

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arXiv2024

Domain walls from SPT-sewing

We introduce a systematic method for constructing gapped domain walls of topologically ordered systems by gauging a lower-dimensional symmetry-protected topological (SPT) order. Based on our construction, we propose a correspondence between 1d SPT phases with a non-invertible $G\times \text{Rep}(G)\times G$ symmetry and invertible domain walls in the quantum double associated with the group $G$. We prove this correspondence when $G$ is Abelian and provide evidence for the general case by studying the quantum double model for $G=S_3$. We also use our method to construct \emph{anchoring domain walls}, which are novel exotic domain walls in the 3d toric code that transform point-like excitations to semi-loop-like excitations anchored on these domain walls.

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arXiv2024

Learning State Preparation Circuits for Quantum Phases of Matter

Many-body ground state preparation is an important subroutine used in the simulation of physical systems. In this paper, we introduce a flexible and efficient framework for obtaining a state preparation circuit for a large class of many-body ground states. We introduce polynomial-time classical algorithms that take reduced density matrices over $\mathcal{O}(1)$-sized balls as inputs, and output a circuit that prepares the global state. We introduce algorithms applicable to (i) short-range entangled states (e.g., states prepared by shallow quantum circuits in any number of dimensions, and more generally, invertible states) and (ii) long-range entangled ground states (e.g., the toric code on a disk). Both algorithms can provably find a circuit whose depth is asymptotically optimal. Our approach uses a variant of the quantum Markov chain condition that remains robust against constant-depth circuits. The robustness of this condition makes our method applicable to a large class of states, whilst ensuring a classically tractable optimization landscape.

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arXiv2024

On the energy barrier of hypergraph product codes

A macroscopic energy barrier is a necessary condition for self-correcting quantum memory. In this paper, we prove tight bounds on the energy barrier applicable to any quantum code obtained from the hypergraph product of two classical codes. If the underlying classical codes are low-density parity-check codes (LDPC), the energy barrier of the quantum code is shown to be the minimum energy barrier of the underlying classical codes (and their transposes) up to an additive $O(1)$ constant.

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arXiv2024

Geometric additivity of modular commutator for multipartite entanglement

A recent surge of research in many-body quantum entanglement has uncovered intriguing properties of quantum many-body systems. A prime example is the modular commutator, which can extract a topological invariant from a single wave function. Here, we unveil novel geometric properties of many-body entanglement via a modular commutator of two-dimensional gapped quantum many-body systems. We obtain the geometric additivity of a modular commutator, indicating that modular commutator for a multipartite system may be an integer multiple of the one for tripartite systems. Using our additivity formula, we also derive a curious identity for the modular commutators involving disconnected intervals in a certain class of conformal field theories. We further illustrate this geometric additivity for both bulk and edge subsystems using numerical calculations of the Haldane and $π$-flux models.

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arXiv2024

Non-unitary Coupled Cluster Enabled by Mid-circuit Measurements on Quantum Computers

Many quantum algorithms rely on a quality initial state for optimal performance. Preparing an initial state for specific applications can considerably reduce the cost of probabilistic algorithms such as the well studied quantum phase estimation (QPE). Fortunately, in the application space of quantum chemistry, generating approximate wave functions for molecular systems is well studied, and quantum computing algorithms stand to benefit from importing these classical methods directly into a quantum circuit. In this work, we propose a state preparation method based on coupled cluster (CC) theory, which is a pillar of quantum chemistry on classical computers, by incorporating mid-circuit measurements into the circuit construction. Currently, the most well studied state preparation method for quantum chemistry on quantum computers is the variational quantum eigensolver (VQE) with a unitary-CC with single- and double-electron excitation terms (UCCSD) ansatz whose operations are limited to unitary gates. We verify the accuracy of our state preparation protocol using mid-circuit measurements by performing energy evaluation and state overlap computation for a set of small chemical systems. We further demonstrate that our approach leads to a reduction of the classical computation overhead, and the number of CNOT and T gates by 28% and 57% on average when compared against the standard VQE-UCCSD protocol.

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