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Bravyi S, Cross AW, Gambetta JM, Maslov D, Rall P, Yoder TJ. High-threshold and low-overhead fault-tolerant quantum memory. Nature 2024; 627:778-782. [PMID: 38538939 PMCID: PMC10972743 DOI: 10.1038/s41586-024-07107-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2023] [Accepted: 01/23/2024] [Indexed: 04/01/2024]
Abstract
The accumulation of physical errors1-3 prevents the execution of large-scale algorithms in current quantum computers. Quantum error correction4 promises a solution by encoding k logical qubits onto a larger number n of physical qubits, such that the physical errors are suppressed enough to allow running a desired computation with tolerable fidelity. Quantum error correction becomes practically realizable once the physical error rate is below a threshold value that depends on the choice of quantum code, syndrome measurement circuit and decoding algorithm5. We present an end-to-end quantum error correction protocol that implements fault-tolerant memory on the basis of a family of low-density parity-check codes6. Our approach achieves an error threshold of 0.7% for the standard circuit-based noise model, on par with the surface code7-10 that for 20 years was the leading code in terms of error threshold. The syndrome measurement cycle for a length-n code in our family requires n ancillary qubits and a depth-8 circuit with CNOT gates, qubit initializations and measurements. The required qubit connectivity is a degree-6 graph composed of two edge-disjoint planar subgraphs. In particular, we show that 12 logical qubits can be preserved for nearly 1 million syndrome cycles using 288 physical qubits in total, assuming the physical error rate of 0.1%, whereas the surface code would require nearly 3,000 physical qubits to achieve said performance. Our findings bring demonstrations of a low-overhead fault-tolerant quantum memory within the reach of near-term quantum processors.
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Affiliation(s)
- Sergey Bravyi
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA
| | - Andrew W Cross
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA
| | - Jay M Gambetta
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA
| | - Dmitri Maslov
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA.
| | - Patrick Rall
- IBM Quantum, MIT-IBM Watson AI Laboratory, Cambridge, MA, USA
| | - Theodore J Yoder
- IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA
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2
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Zhang X, Kim E, Mark DK, Choi S, Painter O. A superconducting quantum simulator based on a photonic-bandgap metamaterial. Science 2023; 379:278-283. [PMID: 36656924 DOI: 10.1126/science.ade7651] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023]
Abstract
Synthesizing many-body quantum systems with various ranges of interactions facilitates the study of quantum chaotic dynamics. Such extended interaction range can be enabled by using nonlocal degrees of freedom such as photonic modes in an otherwise locally connected structure. Here, we present a superconducting quantum simulator in which qubits are connected through an extensible photonic-bandgap metamaterial, thus realizing a one-dimensional Bose-Hubbard model with tunable hopping range and on-site interaction. Using individual site control and readout, we characterize the statistics of measurement outcomes from many-body quench dynamics, which enables in situ Hamiltonian learning. Further, the outcome statistics reveal the effect of increased hopping range, showing the predicted crossover from integrability to ergodicity. Our work enables the study of emergent randomness from chaotic many-body evolution and, more broadly, expands the accessible Hamiltonians for quantum simulation using superconducting circuits.
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Affiliation(s)
- Xueyue Zhang
- Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
| | - Eunjong Kim
- Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
| | - Daniel K Mark
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Soonwon Choi
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Oskar Painter
- Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA.,Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA.,AWS Center for Quantum Computing, Pasadena, CA 91125, USA
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3
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Tremblay MA, Delfosse N, Beverland ME. Constant-Overhead Quantum Error Correction with Thin Planar Connectivity. PHYSICAL REVIEW LETTERS 2022; 129:050504. [PMID: 35960553 DOI: 10.1103/physrevlett.129.050504] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Accepted: 06/15/2022] [Indexed: 06/15/2023]
Abstract
Quantum low density parity check (LDPC) codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, naïve layouts couple many distant qubits with crossing connections which could be hard to build in hardware and could result in performance-degrading crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their Tanner graphs into a small number of planar layers. Each layer contains long-range connections which do not cross. For any Calderbank-Shor-Steane code with a degree-δ Tanner graph, we design stabilizer measurement circuits with depth at most (2δ+2) using at most ⌈δ/2⌉ layers. We observe a circuit-noise threshold of 0.28% for a positive-rate code family using 49 physical qubits per logical qubit. For a physical error rate of 10^{-4}, this family reaches a logical error rate of 10^{-15} using fourteen times fewer physical qubits than the surface code.
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Affiliation(s)
- Maxime A Tremblay
- Institut quantique & Département de physique, Université de Sherbrooke, Sherbrooke, Quebec J1K 2R1, Canada
| | - Nicolas Delfosse
- Microsoft Quantum & Microsoft Research, Redmond, Washington 98052, USA
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4
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Baspin N, Krishna A. Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive. PHYSICAL REVIEW LETTERS 2022; 129:050505. [PMID: 35960571 DOI: 10.1103/physrevlett.129.050505] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2021] [Accepted: 05/26/2022] [Indexed: 06/15/2023]
Abstract
Quantum low-density parity-check (LDPC) codes are a promising avenue to reduce the cost of constructing scalable quantum circuits. However, it is unclear how to implement these codes in practice. Seminal results of Bravyi et al. [Phys. Rev. Lett. 104, 050503 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.050503] have shown that quantum LDPC codes implemented through local interactions obey restrictions on their dimension k and distance d. Here we address the complementary question of how many long-range interactions are required to implement a quantum LDPC code with parameters k and d. In particular, in 2D we show that a quantum LDPC code with distance d∝n^{1/2+ϵ} requires Ω(n^{1/2+ϵ}) interactions of length Ω[over ˜](n^{ϵ}). Further, a code satisfying k∝n with distance d∝n^{α} requires Ω[over ˜](n) interactions of length Ω[over ˜](n^{α/2}). As an application of these results, we consider a model called a stacked architecture, which has previously been considered as a potential way to implement quantum LDPC codes. In this model, although most interactions are local, a few of them are allowed to be very long. We prove that limited long-range connectivity implies quantitative bounds on the distance and code dimension.
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Affiliation(s)
- Nouédyn Baspin
- Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1
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5
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Distance Bounds for Generalized Bicycle Codes. Symmetry (Basel) 2022. [DOI: 10.3390/sym14071348] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
Abstract
Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w.
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6
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Cohen LZ, Kim IH, Bartlett SD, Brown BJ. Low-overhead fault-tolerant quantum computing using long-range connectivity. SCIENCE ADVANCES 2022; 8:eabn1717. [PMID: 35594359 PMCID: PMC10926894 DOI: 10.1126/sciadv.abn1717] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2021] [Accepted: 04/06/2022] [Indexed: 06/15/2023]
Abstract
Vast numbers of qubits will be needed for large-scale quantum computing because of 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 and 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 100 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.
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Affiliation(s)
- Lawrence Z. Cohen
- Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia
| | - Isaac H. Kim
- Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia
- Department of Computer Science, UC Davis, Davis, CA 95616, USA
| | - Stephen D. Bartlett
- Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia
| | - Benjamin J. Brown
- Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia
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7
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Zeng W, Pryadko LP. Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates. PHYSICAL REVIEW LETTERS 2019; 122:230501. [PMID: 31298891 DOI: 10.1103/physrevlett.122.230501] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2018] [Indexed: 06/10/2023]
Abstract
We describe a family of quantum error-correcting codes which generalize both the quantum hypergraph-product codes by Tillich and Zémor and all families of toric codes on m-dimensional hypercubic lattices. Parameters of the constructed codes, including the minimum distances, are given explicitly in terms of those of binary codes associated with the matrices used in the construction.
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Affiliation(s)
- Weilei Zeng
- Department of Physics and Astronomy, University of California, Riverside, California 92521, USA
| | - Leonid P Pryadko
- Department of Physics and Astronomy, University of California, Riverside, California 92521, USA
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8
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Campbell ET, Terhal BM, Vuillot C. Roads towards fault-tolerant universal quantum computation. Nature 2017; 549:172-179. [DOI: 10.1038/nature23460] [Citation(s) in RCA: 207] [Impact Index Per Article: 29.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2017] [Accepted: 04/28/2017] [Indexed: 12/20/2022]
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9
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Bolt A, Duclos-Cianci G, Poulin D, Stace TM. Foliated Quantum Error-Correcting Codes. PHYSICAL REVIEW LETTERS 2016; 117:070501. [PMID: 27563942 DOI: 10.1103/physrevlett.117.070501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2016] [Indexed: 06/06/2023]
Abstract
We show how to construct a large class of quantum error-correcting codes, known as Calderbank-Steane-Shor codes, from highly entangled cluster states. This becomes a primitive in a protocol that foliates a series of such cluster states into a much larger cluster state, implementing foliated quantum error correction. We exemplify this construction with several familiar quantum error-correction codes and propose a generic method for decoding foliated codes. We numerically evaluate the error-correction performance of a family of finite-rate Calderbank-Steane-Shor codes known as turbo codes, finding that they perform well over moderate depth foliations. Foliated codes have applications for quantum repeaters and fault-tolerant measurement-based quantum computation.
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Affiliation(s)
- A Bolt
- ARC Centre for Engineered Quantum System, Department of Physics, University of Queensland, Brisbane, Queensland 4072, Australia
| | - G Duclos-Cianci
- Département de Physique, Université de Sherbrooke, Québec J1K 2R1, Canada
| | - D Poulin
- Département de Physique, Université de Sherbrooke, Québec J1K 2R1, Canada
| | - T M Stace
- ARC Centre for Engineered Quantum System, Department of Physics, University of Queensland, Brisbane, Queensland 4072, Australia
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10
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Dumer I, Kovalev AA, Pryadko LP. Thresholds for Correcting Errors, Erasures, and Faulty Syndrome Measurements in Degenerate Quantum Codes. PHYSICAL REVIEW LETTERS 2015; 115:050502. [PMID: 26274403 DOI: 10.1103/physrevlett.115.050502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2014] [Indexed: 06/04/2023]
Abstract
We suggest a technique for constructing lower (existence) bounds for the fault-tolerant threshold to scalable quantum computation applicable to degenerate quantum codes with sublinear distance scaling. We give explicit analytic expressions combining probabilities of erasures, depolarizing errors, and phenomenological syndrome measurement errors for quantum low-density parity-check codes with logarithmic or larger distances. These threshold estimates are parametrically better than the existing analytical bound based on percolation.
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Affiliation(s)
- Ilya Dumer
- Department of Electrical Engineering, University of California, Riverside, California 92521, USA
| | - Alexey A Kovalev
- Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA
| | - Leonid P Pryadko
- Department of Physics and Astronomy, University of California, Riverside, California 92521, USA
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11
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Marvian I, Lidar DA. Quantum error suppression with commuting Hamiltonians: two local is too local. PHYSICAL REVIEW LETTERS 2014; 113:260504. [PMID: 25615294 DOI: 10.1103/physrevlett.113.260504] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2014] [Indexed: 06/04/2023]
Abstract
We consider error suppression schemes in which quantum information is encoded into the ground subspace of a Hamiltonian comprising a sum of commuting terms. Since such Hamiltonians are gapped, they are considered natural candidates for protection of quantum information and topological or adiabatic quantum computation. However, we prove that they cannot be used to this end in the two-local case. By making the favorable assumption that the gap is infinite, we show that single-site perturbations can generate a degeneracy splitting in the ground subspace of this type of Hamiltonian which is of the same order as the magnitude of the perturbation, and is independent of the number of interacting sites and their Hilbert space dimensions, just as in the absence of the protecting Hamiltonian. This splitting results in decoherence of the ground subspace, and we demonstrate that for natural noise models the coherence time is proportional to the inverse of the degeneracy splitting. Our proof involves a new version of the no-hiding theorem which shows that quantum information cannot be approximately hidden in the correlations between two quantum systems. The main reason that two-local commuting Hamiltonians cannot be used for quantum error suppression is that their ground subspaces have only short-range (two-body) entanglement.
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Affiliation(s)
- Iman Marvian
- Center for Quantum Information Science and and Technology, University of Southern California, Los Angeles, California 90089, USA and Department of Physics, University of Southern California, Los Angeles, California 90089, USA
| | - Daniel A Lidar
- Center for Quantum Information Science and and Technology, University of Southern California, Los Angeles, California 90089, USA and Department of Physics, University of Southern California, Los Angeles, California 90089, USA and Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA and Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA
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12
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Kim IH. Long-range entanglement is necessary for a topological storage of quantum information. PHYSICAL REVIEW LETTERS 2013; 111:080503. [PMID: 24010422 DOI: 10.1103/physrevlett.111.080503] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/24/2013] [Revised: 07/31/2013] [Indexed: 06/02/2023]
Abstract
A general inequality between entanglement entropy and a number of topologically ordered states is derived, even without using the properties of the parent Hamiltonian or the formalism of topological quantum field theory. Given a quantum state |ψ], we obtain an upper bound on the number of distinct states that are locally indistinguishable from |ψ]. The upper bound is determined only by the entanglement entropy of some local subsystems. As an example, we show that log N≤2γ for a large class of topologically ordered systems on a torus, where N is the number of topologically protected states and γ is the constant subcorrection term of the entanglement entropy. We discuss applications to quantum many-body systems that do not have any low-energy topological quantum field theory description, as well as tradeoff bounds for general quantum error correcting codes.
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Affiliation(s)
- Isaac H Kim
- Institute of Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
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13
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Bravyi S, König R. Classification of topologically protected gates for local stabilizer codes. PHYSICAL REVIEW LETTERS 2013; 110:170503. [PMID: 23679695 DOI: 10.1103/physrevlett.110.170503] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2012] [Indexed: 06/02/2023]
Abstract
Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological protection must be "turned off" for at least some steps in the computation in order to achieve universality. For the 3D geometry we show that an encoded gate U is implementable by a constant-depth circuit only if UPU(†) is in the Clifford group for any Pauli operator P. This class of gates includes some non-Clifford gates such as the π/8 rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance.
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Affiliation(s)
- Sergey Bravyi
- IBM Watson Research Center, Yorktown Heights, New York 10598, USA
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14
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Landon-Cardinal O, Poulin D. Local topological order inhibits thermal stability in 2D. PHYSICAL REVIEW LETTERS 2013; 110:090502. [PMID: 23496697 DOI: 10.1103/physrevlett.110.090502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2012] [Indexed: 06/01/2023]
Abstract
We study the robustness of quantum information stored in the degenerate ground space of a local, frustration-free Hamiltonian with commuting terms on a 2D spin lattice. On one hand, a macroscopic energy barrier separating the distinct ground states under local transformations would protect the information from thermal fluctuations. On the other hand, local topological order would shield the ground space from static perturbations. Here we demonstrate that local topological order implies a constant energy barrier, thus inhibiting thermal stability.
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15
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Hastings MB. Topological order at nonzero temperature. PHYSICAL REVIEW LETTERS 2011; 107:210501. [PMID: 22181865 DOI: 10.1103/physrevlett.107.210501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2011] [Indexed: 05/31/2023]
Abstract
We propose a definition for topological order at nonzero temperature in analogy to the usual zero temperature definition that a state is topologically ordered, or "nontrivial", if it cannot be transformed into a product state (or a state close to a product state) using a local (or approximately local) quantum circuit. We prove that any two-dimensional Hamiltonian which is a sum of commuting local terms is not topologically ordered at T > 0. We show that such trivial states cannot be used to store quantum information using certain stringlike operators. This definition is not too restrictive, however, as the four dimensional toric code does have a nontrivial phase at nonzero temperature.
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Affiliation(s)
- Matthew B Hastings
- Duke University, Department of Physics, Durham, North Carolina, 27708, USA
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