Publications (72)287.12 Total impact
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ABSTRACT: We analyse a model for faulttolerant quantum computation with low overhead suitable for situations where the noise is biased. The basis for this scheme is a gadget for the faulttolerant preparation of magic states that enable universal faulttolerant quantum computation using only Clifford gates that preserve the noise bias. We analyse the distillation of $T\rangle$type magic states using this gadget at the physical level, followed by concatenation with the 15qubit quantum ReedMuller code, and comparing our results with standard constructions. In the regime where the noise bias (rate of Pauli $Z$ errors relative to other singlequbit errors) is greater than a factor of 10, our scheme has lower overhead across a broad range of relevant noise rates.  [Show abstract] [Hide abstract]
ABSTRACT: A twodimensional topologically ordered quantum memory is well protected against error if the energy gap is large compared to the temperature, but this protection does not improve as the system size increases. We review and critique some recent proposals for improving the memory time by introducing longrange interactions among anyons, noting that instability with respect to small local perturbations of the Hamiltonian is a generic problem for such proposals. We also discuss some broader issues regarding the prospects for scalable quantum memory in twodimensional systems.  [Show abstract] [Hide abstract]
ABSTRACT: A twodimensional topologically ordered quantum memory is well protected against error if the energy gap is large compared to the temperature, but this protection does not improve as the system size increases. We review and critique some recent proposals for improving the memory time by introducing longrange interactions among anyons, noting that instability with respect to small local perturbations of the Hamiltonian is a generic problem for such proposals. We also discuss some broader issues regarding the prospects for scalable quantum memory in twodimensional systems.  [Show abstract] [Hide abstract]
ABSTRACT: The simulation of molecules is a widely anticipated application of quantum computers. However, recent studies \cite{WBCH13a,HWBT14a} have cast a shadow on this hope by revealing that the complexity in gate count of such simulations increases with the number of spin orbitals $N$ as $N^8$, which becomes prohibitive even for molecules of modest size $N\sim 100$. This study was partly based on a scaling analysis of the Trotter step required for an ensemble of random artificial molecules. Here, we revisit this analysis and find instead that the scaling is closer to $N^6$ in worst case for real model molecules we have studied, indicating that the random ensemble fails to accurately capture the statistical properties of realworld molecules. Actual scaling may be significantly better than this due to averaging effects. We then present an alternative simulation scheme and show that it can sometimes outperform existing schemes, but that this possibility depends crucially on the details of the simulated molecule. We obtain further improvements using a version of the coalescing scheme of \cite{WBCH13a}; this scheme is based on using different Trotter steps for different terms. The method we use to bound the complexity of simulating a given molecule is efficient, in contrast to the approach of \cite{WBCH13a,HWBT14a} which relied on exponentially costly classical exact simulation.  [Show abstract] [Hide abstract]
ABSTRACT: We introduce a new class of circuits for constructing efficiently decodable quantum and classical errorcorrection codes, based on a recently discovered contractible tensor network known as branching multiscale entanglement renormalization ansatz [1]. We perform an indepth study of a particular example that can be thought of as an extension to Arikan's polar code [2][4]. Notably, our numerical simulation show that these codes polarize the logical channels more strongly while retaining the loglinear decoding complexity using the successive cancellation decoder. These codes also display improved errorcorrecting capability with only a minor impact on decoding complexity. Efficient decoding is realized using powerful graphical calculus tools developed in the field of quantum manybody physics.  [Show abstract] [Hide abstract]
ABSTRACT: In leading faulttolerant quantum computing schemes, accurate transformation are obtained by a twostage process. In a first stage, a discrete, universal set of faulttolerant operations is obtained by errorcorrecting noisy transformations and distilling resource states. In a second stage, arbitrary transformations are synthesized to desired accuracy by combining elements of this set into a circuit. Here, we present a scheme which merges these two stages into a single one, directly distilling complex transformations. We find that our scheme can reduce the total overhead to realize certain gates by up to a few orders of magnitude. In contrast to other schemes, this efficient gate synthesis does not require computationally intensive compilation algorithms, and a straightforward generalization of our scheme circumvents compilation and synthesis altogether.  [Show abstract] [Hide abstract]
ABSTRACT: Steane's 7qubit quantum errorcorrecting code admits a set of faulttolerant gates that generate the Clifford group, which in itself is not universal for quantum computation. The 15qubit ReedMuller code also does not admit a universal faulttolerant gate set but possesses faulttolerant T and controlcontrolZ gates. Combined with the Clifford group, either of these two gates generate a universal set. Here, we combine these two features by demonstrating how to faulttolerantly convert between these two codes, providing a new method to realize universal faulttolerant quantum computation. One interpretation of our result is that both codes correspond to the same subsystem code in different gauges. Our scheme extends to the entire family of quantum ReedMuller codes.  [Show abstract] [Hide abstract]
ABSTRACT: We introduce a new class of circuits for constructing efficiently decodable errorcorrection codes, based on a recently discovered contractible tensor network. We perform an indepth study of a particular example that can be thought of as an extension to Arikan's polar code. Notably, our numerical simulation show that this code polarizes the logical channels more strongly while retaining the loglinear decoding complexity using the successive cancellation decoder. These codes also display improved errorcorrecting capability with only a minor impact on decoding complexity. Efficient decoding is realized using powerful graphical calculus tools developed in the field of quantum manybody physics. In a companion paper, we generalize our construction to the quantum setting and describe more indepth the relation between classical and quantum error correction and the graphical calculus of tensor networks.  [Show abstract] [Hide abstract]
ABSTRACT: We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum manybody physics. We exhibit correspondences between wellknown families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.  [Show abstract] [Hide abstract]
ABSTRACT: We consider twodimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath. We propose a phenomenological model for the resulting shorttime dynamics that includes paircreation, hopping, braiding, and fusion of anyons. By explicitly constructing topological quantum errorcorrecting codes for this class of system, we use our thermalization model to estimate the lifetime of the quantum information stored in the encoded spaces. To decode and correct errors in these codes, we adapt several existing topological decoders to the nonAbelian setting. We perform largescale numerical simulations of these twodimensional Ising anyon systems and find that the thresholds of these models range between 13% to 25%. To our knowledge, these are the first numerical threshold estimates for quantum codes without explicit additive structure.  [Show abstract] [Hide abstract]
ABSTRACT: In this article we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the decoding problem consists of determining the most likely recovery given the syndrome. The corresponding classical problem is known to be NPcomplete, and a similar decoding problem for quantum codes is also known to be NPcomplete. However, this decoding strategy is not optimal in the quantum setting as it does not take into account error degeneracy, which causes distinct errors to have the same effect on the code. Here, we show that optimal decoding of stabilizer codes is computationally much harder than optimal decoding of classical linear codes, it is #P. 
Article: Kitaev's Zdcode threshold estimates
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ABSTRACT: We study the quantum error correction threshold of Kitaev's toric code over the group Zd subject to a generalized bitflip noise. This problem requires special decoding techniques, and for this purpose we generalize the renormalizationgroup method we introduced previously [ G. DuclosCianci and D. Poulin Phys. Rev. Lett. 104 050504 (2010) and IEEE Information Theory Workshop, Dublin (2010), p. 1] for Z2 topological codes.  [Show abstract] [Hide abstract]
ABSTRACT: We present a threedimensional generalization of a renormalization group decoding algorithm for topological codes with Abelian anyonic excitations that we previously introduced for two dimensions. This 3D implementation extends our previous 2D algorithm by incorporating a failure probability of the syndrome measurements, i.e., it enables faulttolerant decoding. We report a faulttolerant storage threshold of 1.9(4)% for Kitaev's toric code subject to a 3D bitflip channel (i.e. including imperfect syndrome measurements). This number is to be compared with the 2.9% value obtained via perfect matching. The 3D generalization inherits many properties of the 2D algorithm, including a complexity linear in the spacetime volume of the memory, which can be parallelized to logarithmic time.  [Show abstract] [Hide abstract]
ABSTRACT: We study the robustness of quantum information stored in the degenerate ground space of a local, frustrationfree 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.  [Show abstract] [Hide abstract]
ABSTRACT: Studying large manybody quantum systems is difficult because the dimension of the Hilbert space grows exponentially with the number of particles/subsystems. I will present a method to approximately calculate the finitetemperature properties of an infinite, translationally invariant system by just keeping knowledge of small, local subsystems. Key to this method is the ability to (over)estimate the global entropy, giving us access to the Gibbs free energy, and results in the property that we can find a rigorous lowerbound to the ground state energy (which compliments rigorous upperbounds that can be found with more common, variational techniques).  [Show abstract] [Hide abstract]
ABSTRACT: The Markov entropy decomposition (MED) is a recentlyproposed, clusterbased simulation method for finite temperature quantum systems with arbitrary geometry. In this paper, we detail numerical algorithms for performing the required steps of the MED, principally solving a minimization problem with a preconditioned Newton's algorithm, as well as how to extract global susceptibilities and thermal responses. We demonstrate the power of the method with the spin1/2 XXZ model on the 2D square lattice, including the extraction of critical points and details of each phase. Although the method shares some qualitative similarities with exactdiagonalization, we show the MED is both more accurate and significantly more flexible.  [Show abstract] [Hide abstract]
ABSTRACT: Topological phases can be defined in terms of local equivalence: two systems are in the same topological phase if it is possible to transform one into the other by a local reorganization of its degrees of freedom. The classification of topological phases therefore amounts to the classification of longrange entanglement. Such local transformation could result, for instance, from the adiabatic continuation of one system's Hamiltonian to the other. Here, we use this definition to study the topological phase of translationally invariant stabilizer codes in two spatial dimensions, and show that they all belong to one universal phase. We do this by constructing an explicit mapping from any such code to a number of copies of Kitaev's code. Some of our results extend to some twodimensional (2D) subsystem codes, including topological subsystem codes. Error correction benefits from the corresponding local mappings. In particular, it enables us to use decoding algorithm developed for Kitaev's code to decode any 2D stabilizer code and subsystem code.  [Show abstract] [Hide abstract]
ABSTRACT: We propose a simplified version of the Kitaev's surface code in which error correction requires only threequbit parity measurements for Pauli operators XXX and ZZZ. The new code belongs to the class of subsystem stabilizer codes. It inherits many favorable properties of the standard surface code such as encoding of multiple logical qubits on a planar lattice with punctured holes, efficient decoding by either minimumweight matching or renormalization group methods, and high error threshold. The new subsystem surface code (SSC) gives rise to an exactly solvable Hamiltonian with 3qubit interactions, topologically ordered ground state, and a constant energy gap. We construct a local unitary transformation mapping the SSC Hamiltonian to the one of the ordinary surface code thus showing that the two Hamiltonians belong to the same topological class. We describe error correction protocols for the SSC and determine its error thresholds under several natural error models. In particular, we show that the SSC has error threshold approximately 0.6% for the standard circuitbased error model studied in the literature. We also consider a model in which threequbit parity operators can be measured directly. We show that the SSC has error threshold approximately 0.97% in this setting.  [Show abstract] [Hide abstract]
ABSTRACT: Quantum Markov networks are a generalization of quantum Markov chains to arbitrary graphs. They provide a powerful classification of correlations in quantum manybody systemscomplementing the area law at finite temperatureand are therefore useful to understand the powers and limitations of certain classes of simulation algorithms. Here, we extend the characterization of quantum Markov networks and in particular prove the equivalence of positive quantum Markov networks and Gibbs states of Hamiltonians that are the sum of local commuting terms on graphs containing no triangles. For more general graphs we demonstrate the equivalence between quantum Markov networks and Gibbs states of a class of Hamiltonians of intermediate complexity between commuting and general local Hamiltonians.  [Show abstract] [Hide abstract]
ABSTRACT: We present an experimental procedure to determine the usefulness of a measurement scheme for quantum error correction (QEC). A QEC scheme typically requires the ability to prepare entangled states, to carry out multiqubit measurements, and to perform certain recovery operations conditioned on measurement outcomes. As a consequence, the experimental benchmark of a QEC scheme is a tall order because it requires the conjuncture of many elementary components. Our scheme opens the path to experimental benchmarks of individual components of QEC. Our numerical simulations show that certain parity measurements realized in circuit quantum electrodynamics are on the verge of being useful for QEC.
Publication Stats
2k  Citations  
287.12  Total Impact Points  
Top Journals
Institutions

20092015

Université de Sherbrooke
 Department of Physics
Шербрук, Quebec, Canada


20062008

California Institute of Technology
 Institute for Quantum Information and Matter
Pasadena, California, United States


20062007

University of Queensland
Brisbane, Queensland, Australia


20032005

University of Waterloo
 Institute for Quantum Computing
Waterloo, Ontario, Canada 
Perimeter Institute for Theoretical Physics
Waterloo, Ontario, Canada


2002

Los Alamos National Laboratory
 Theoretical Division
ЛосАламос, California, United States


2001

Université de Montréal
Montréal, Quebec, Canada
