arXiv subjects

Isaac H. Kim

Publications and source records attributed to Isaac H. Kim.

At least 37 records · Page 2Linked to original sources

arXiv2024

Classifying 2D topological phases: mapping ground states to string-nets

We prove the conjectured classification of topological phases in two spatial dimensions with gappable boundary, in a simplified setting. Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth quantum circuit. It is conjectured that the Levin-Wen string-net models exhaust all possible gapped phases with gappable boundary, and these phases are labeled by unitary modular tensor categories. We prove this under the assumption that every phase has a representative state with zero correlation length satisfying the entanglement bootstrap axioms, or a strict form of area law. Our main technical development is to transform these states into string-net states using constant-depth quantum circuits.

quant-ph↗

arXiv2024

The Cost of Entanglement Renormalization on a Fault-Tolerant Quantum Computer

We perform a detailed resource estimate for the prospect of using deep entanglement renormalization ansatz (DMERA) on a fault-tolerant quantum computer, focusing on the regime in which the target system is large. For probing a relatively large system size ($64\times 64$), we observe up to an order of magnitude reduction in the number of qubits, compared to the approaches based on quantum phase estimation (QPE). We discuss two complementary strategies to measure the energy. The first approach is based on a random sampling of the local terms of the Hamiltonian, requiring $\mathcal{O}(1/ε^2)$ invocations of quantum circuits, each of which have depth of at most $\mathcal{O}(\log N)$, where $ε$ is the relative precision in the energy and $N$ is the system size. The second approach is based on a coherent estimation of the expectation value of observables averaged over space, which achieves the Heisenberg scaling while incurring only a logarithmic cost in the system size. For estimating the energy per site of $ε$, $\mathcal{O}\left(\frac{\log N}ε \right)$ $T$ gates and $\mathcal{O}\left(\log N \right)$ qubits suffice. The constant factor of the leading contribution is shown to be determined by the depth of the DMERA circuit, the gates used in the ansatz, and the periodicity of the circuit. We also derive tight bounds on the variance of the energy gradient, assuming the gates are random Pauli rotations.

quant-ph↗

arXiv2024

Strict area law implies commuting parent Hamiltonian

We show that in two spatial dimensions, when a quantum state has entanglement entropy obeying a strict area law, meaning $S(A)=α|\partial A| - γ$ for constants $α, γ$ independent of lattice region $A$, then it admits a commuting parent Hamiltonian. More generally, we prove that the entanglement bootstrap axioms in 2D imply the existence of a commuting, local parent Hamiltonian with a stable spectral gap. We also extend our proof to states that describe gapped domain walls. Physically, these results imply that the states studied in the entanglement bootstrap program correspond to ground states of some local Hamiltonian, describing a stable phase of matter. Our result also suggests that systems with chiral gapless edge modes cannot obey a strict area law provided they have finite local Hilbert space.

quant-ph↗

arXiv2023

Universal lower bound on topological entanglement entropy

Entanglement entropies of two-dimensional gapped ground states are expected to satisfy an area law, with a constant correction term known as the topological entanglement entropy (TEE). In many models, the TEE takes a universal value that characterizes the underlying topological phase. However, the TEE is not truly universal: it can differ even for two states related by constant-depth circuits, which are necessarily in the same phase. The difference between the TEE and the value predicted by the anyon theory is often called the spurious topological entanglement entropy. We show that this spurious contribution is always nonnegative, thus the value predicted by the anyon theory provides a universal lower bound. This observation also leads to a definition of TEE that is invariant under constant-depth quantum circuits.

quant-ph↗

arXiv2023

Qubit recycling and the path counting problem

Recently, it was shown that the qudits used in circuits of a convolutional form (e.g., Matrix Product State sand Multi-scale Entanglement Renormalization Ansatz) can be reset unitarily \href{https://doi.org/10.1103/PhysRevA.103.042613}{[Phys. Rev. A 103, 042613 (2021)]}, even without measurement. We analyze the fidelity of this protocol for a family of quantum circuits that interpolates between such circuits and local quantum circuits, averaged over Haar-random gates. We establish a connection between this problem and a counting of directed paths on a graph, which is determined by the shape of the quantum circuit. This connection leads to an exact expression for the fidelity of the protocol for the entire family that interpolates between convolutional circuit and random quantum circuit. For convolutional circuits of constant window size, the rate of convergence to unit fidelity is shown to be $\frac{q^2}{q^2+1}$, independent of the window size, where $q$ is the local qudit dimension. Since most applications of convolutional circuits use constant-sized windows, our result suggests that the unitary reset protocol will likely work well in such a regime. We also derive two extra results in the convolutional limit, which may be of an independent interest. First, we derive exact expressions for the correlations between reset qudits and show that it decays exponentially in the distance. Second, we derive an expression for the the fidelity in the presence of noise, expressed in terms of the quantities that define the property of the channel, such as the entanglement fidelity.

quant-ph↗

arXiv2023

A new operator extension of strong subadditivity of quantum entropy

Let $S(ρ)$ be the von Neumann entropy of a density matrix $ρ$. Weak monotonicity asserts that $S(ρ_{AB}) - S(ρ_A) + S(ρ_{BC}) - S(ρ_C)\geq 0$ for any tripartite density matrix $ρ_{ABC}$, a fact that is equivalent to the strong subadditivity of entropy. We prove an operator inequality, which, upon taking an expectation value with respect to the state $ρ_{ABC}$, reduces to the weak monotonicity inequality. Generalizations of this inequality to the one involving two independent density matrices, as well as their Rényi-generalizations, are also presented.

quant-ph↗

arXiv2023

Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules

We estimate the resources required in the fusion-based quantum computing scheme to simulate electrolyte molecules in Li-ion batteries on a fault-tolerant, photonic quantum computer. We focus on the molecules that can provide practical solutions to industrially relevant problems. Certain fault-tolerant operations require the use of single-qubit "magic states" prepared by dedicated "magic state factories" (MSFs). Producing and consuming magic states in parallel is typically a prohibitively expensive task, resulting in the serial application of fault-tolerant gates. However, for the systems considered, the MSF constitutes a negligible fraction of the total footprint of the quantum computer, allowing for the use of multiple MSFs to produce magic states in parallel. We suggest architectural and algorithmic techniques that can accommodate such a capability. We propose a method to consume multiple magic states simultaneously, which can potentially lead to an order of magnitude reduction in the computational runtime without additional expense in the footprint.

quant-ph↗

arXiv2023

Measurement as a shortcut to long-range entangled quantum matter

The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback (``adaptive circuits'') can evade such restrictions. We introduce three classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs). The three classes are inspired by distinct physical insights, including tensor-network constructions, multiscale entanglement renormalization ansatz (MERA), and parton constructions. A large class of topological orders, including chiral topological order, can be prepared in constant depth or time, and one-dimensional CFT states and non-abelian topological orders with both solvable and non-solvable groups can be prepared in depth scaling logarithmically with system size. We also build on a recently discovered correspondence between symmetry-protected topological phases and long-range entanglement to derive efficient protocols for preparing symmetry-enriched topological order and arbitrary CSS (Calderbank-Shor-Steane) codes. Our work illustrates the practical and conceptual versatility of measurement for state preparation.

cond-mat.str-el↗

arXiv2023

Complementarity and the unitarity of the black hole $S$-matrix

Recently, Akers et al. proposed a non-isometric holographic map from the interior of a black hole to its exterior. Within this model, we study properties of the black hole $S$-matrix, which are in principle accessible to observers who stay outside the black hole. Specifically, we investigate a scenario in which an infalling agent interacts with radiation both outside and inside the black hole. Because the holographic map involves postselection, the unitarity of the $S$-matrix is not guaranteed in this scenario, but we find that unitarity is satisfied to very high precision if suitable conditions are met. If the internal black hole dynamics is described by a pseudorandom unitary transformation, and if the operations performed by the infaller have computational complexity scaling polynomially with the black hole entropy, then the $S$-matrix is unitary up to corrections that are superpolynomially small in the black hole entropy. Furthermore, while in principle quantum computation assisted by postselection can be very powerful, we find under similar assumptions that the $S$-matrix of an evaporating black hole has polynomial computational complexity.

hep-th↗

arXiv2023

Experimental demonstration of the advantage of adaptive quantum circuits

Adaptive quantum circuits employ unitary gates assisted by mid-circuit measurement, classical computation on the measurement outcome, and the conditional application of future unitary gates based on the result of the classical computation. In this paper, we experimentally demonstrate that even a noisy adaptive quantum circuit of constant depth can achieve a task that is impossible for any purely unitary quantum circuit of identical depth: the preparation of long-range entangled topological states with high fidelity. We prepare a particular toric code ground state with fidelity of at least $76.9\pm 1.3\%$ using a constant depth ($d=4$) adaptive circuit, and rigorously show that no unitary circuit of the same depth and connectivity could prepare this state with fidelity greater than $50\%$.

quant-ph↗

arXiv2022

Circuit depth versus energy in topologically ordered systems

We prove a nontrivial circuit-depth lower bound for preparing a low-energy state of a locally interacting quantum many-body system in two dimensions, assuming the circuit is geometrically local. For preparing any state which has an energy density of at most $ε$ with respect to Kitaev's toric code Hamiltonian on a two dimensional lattice $Λ$, we prove a lower bound of $Ω\left(\min\left(1/ε^{\frac{1-α}{2}}, \sqrt{|Λ|}\right)\right)$ for any $α>0$. We discuss two implications. First, our bound implies that the lowest energy density obtainable from a large class of existing variational circuits (e.g., Hamiltonian variational ansatz) cannot, in general, decay exponentially with the circuit depth. Second, if long-range entanglement is present in the ground state, this can lead to a nontrivial circuit-depth lower bound even at nonzero energy density. Unlike previous approaches to prove circuit-depth lower bounds for preparing low energy states, our proof technique does not rely on the ground state to be degenerate.

quant-ph↗

arXiv2022

Modular Commutators in Conformal Field Theory

The modular commutator is a recently discovered multipartite entanglement measure that quantifies the chirality of the underlying many-body quantum state. In this Letter, we derive a universal expression for the modular commutator in conformal field theories in $1+1$ dimensions and discuss its salient features. We show that the modular commutator depends only on the chiral central charge and the conformal cross ratio. We test this formula for a gapped $(2+1)$-dimensional system with a chiral edge, i.e., the quantum Hall state, and observe excellent agreement with numerical simulations. Furthermore, we propose a geometric dual for the modular commutator in certain preferred states of the AdS/CFT correspondence. For these states, we argue that the modular commutator can be obtained from a set of crossing angles between intersecting Ryu-Takayanagi surfaces.

cond-mat.str-el↗

arXiv2022

Low-overhead fault-tolerant quantum computing using long-range connectivity

Vast numbers of qubits will be needed for large-scale quantum computing due to the overheads associated with error correction. We present a scheme for low-overhead fault-tolerant quantum computation based on quantum low-density parity-check (LDPC) codes, where long-range interactions enable many logical qubits to be encoded with a modest number of physical qubits. In our approach, logic gates operate via logical Pauli measurements that preserve both the protection of the LDPC codes as well as the low overheads in terms of the required number of additional qubits. Compared with surface codes with the same code distance, we estimate order-of-magnitude improvements in the overheads for processing around one hundred logical qubits using this approach. Given the high thresholds demonstrated by LDPC codes, our estimates suggest that fault-tolerant quantum computation at this scale may be achievable with a few thousand physical qubits at comparable error rates to what is needed for current approaches.

quant-ph↗

arXiv2022

Fast digital methods for adiabatic state preparation

We present a quantum algorithm for adiabatic state preparation on a gate-based quantum computer, with complexity polylogarithmic in the inverse error. Our algorithm digitally simulates the adiabatic evolution between two self-adjoint operators $H_0$ and $H_1$, exponentially suppressing the diabatic error by harnessing the theoretical concept of quasi-adiabatic continuation as an algorithmic tool. Given an upper bound $α$ on $\|H_0\|$ and $\|H_1\|$ along with the promise that the $k$th eigenstate $|ψ_k(s)\rangle$ of $H(s) \equiv (1-s)H_0 + sH_1$ is separated from the rest of the spectrum by a gap of at least $γ> 0$ for all $s \in [0,1]$, this algorithm implements an operator $\widetilde{U}$ such that $\||ψ_k(1)\rangle - \widetilde{U}|ψ_k(s)\rangle\| \leq ε$ using $O(α^2/γ^2)\text{polylog}(α/γε)$ queries to block-encodings of $H_0$ and $H_1$. In addition, we develop an algorithm that is applicable only to ground states and requires multiple queries to an oracle that prepares $|ψ_0(0)\rangle$, but has slightly better scaling in all parameters. We also show that the costs of both algorithms can be further reduced under certain reasonable conditions, such as when $\|H_1 - H_0\|$ is small compared to $α$, or when more information about the gap of $H(s)$ is available. For certain problems, the scaling can even be improved to linear in $\|H_1 - H_0\|/γ$ up to polylogarithmic factors.

quant-ph↗

arXiv2021

Fault-tolerant qubit from a constant number of components

With gate error rates in multiple technologies now below the threshold required for fault-tolerant quantum computation, the major remaining obstacle to useful quantum computation is scaling, a challenge greatly amplified by the huge overhead imposed by quantum error correction itself. We propose a fault-tolerant quantum computing scheme that can nonetheless be assembled from a small number of experimental components, potentially dramatically reducing the engineering challenges associated with building a large-scale fault-tolerant quantum computer. Our scheme has a threshold of 0.39% for depolarising noise, assuming that memory errors are negligible. In the presence of memory errors, the logical error rate decays exponentially with $\sqrt{T/τ}$, where $T$ is the memory coherence time and $τ$ is the timescale for elementary gates. Our approach is based on a novel procedure for fault-tolerantly preparing three-dimensional cluster states using a single actively controlled qubit and a pair of delay lines. Although a circuit-level error may propagate to a high-weight error, the effect of this error on the prepared state is always equivalent to that of a constant-weight error. We describe how the requisite gates can be implemented using existing technologies in quantum photonic and phononic systems. With continued improvements in only a few components, we expect these systems to be promising candidates for demonstrating fault-tolerant quantum computation with a comparatively modest experimental effort.

quant-ph↗

arXiv2021

Limits on the storage of quantum information in a volume of space

We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits $n$, the number of encoded qubits $k$, the code distance $d$, the accuracy parameter $δ$ that quantifies how well the erasure channel can be reversed, and the locality parameter $\ell$ that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy $ε$ that is exponentially small in $\ell$, which is the case for perturbations of local commuting projector codes, our bound reads $kd^{\frac{2}{D-1}} \le O\bigl(n (\log n)^{\frac{2D}{D-1}} \bigr)$ for codes on $D$-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed $O\bigl(\ell n^{(D-1)/D}\bigr)$ if $δ\le O(\ell n^{-1/D})$. As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance $d$ and the size $\tilde d$ of a minimal region that can support all approximate logical operators satisfies $\tilde d d^{\frac{1}{D-1}}\le O\bigl( n \ell^{\frac{D}{D-1}} \bigr)$, where the logical operators are accurate up to $O\bigl( ( n δ/ d )^{1/2}\bigr)$ in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.

quant-ph↗

arXiv2021

Modular commutator in gapped quantum many-body systems

In arXiv:2110.06932, we argued that the chiral central charge -- a topologically protected quantity characterizing the edge theory of a gapped (2+1)-dimensional system -- can be extracted from the bulk by using an order parameter called the modular commutator. In this paper, we reveal general properties of the modular commutator and strengthen its relationship with the chiral central charge. First, we identify connections between the modular commutator and conditional mutual information, time reversal, and modular flow. Second, we prove, within the framework of the entanglement bootstrap program, that two topologically ordered media connected by a gapped domain wall must have the same modular commutator in their respective bulk. Third, we numerically calculate the value of the modular commutator for a bosonic lattice Laughlin state for finite sizes and extrapolate to the infinite-volume limit. The result of this extrapolation is consistent with the proposed formula up to an error of about 0.7%.

quant-ph↗

arXiv2021

Chiral central charge from a single bulk wave function

A $(2+1)$-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real-valued in any local product basis.

quant-ph↗