Publications (17)73.73 Total impact
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Article: Fault-Tolerant Renormalization Group Decoder for Abelian Topological Codes
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ABSTRACT: We present a three-dimensional 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 fault-tolerant decoding. We report a fault-tolerant storage threshold of 1.9(4)% for Kitaev's toric code subject to a 3D bit-flip 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 space-time volume of the memory, which can be parallelized to logarithmic time.04/2013; -
Article: Local Topological Order Inhibits Thermal Stability in 2D.
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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.Physical Review Letters 03/2013; 110(9):090502. · 7.37 Impact Factor -
Article: Kitaev's Z_d-Codes Threshold Estimates
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ABSTRACT: We study the quantum error correction threshold of Kitaev's toric code over the group Z_d subject to a generalized bit-flip noise. This problem requires novel decoding techniques, and for this purpose we generalize the renormalization group method we previously introduced for Z_2 topological codes.02/2013; -
Article: Subsystem surface codes with three-qubit check operators
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ABSTRACT: We propose a simplified version of the Kitaev's surface code in which error correction requires only three-qubit 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 minimum-weight matching or renormalization group methods, and high error threshold. The new subsystem surface code (SSC) gives rise to an exactly solvable Hamiltonian with 3-qubit 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 circuit-based error model studied in the literature. We also consider a model in which three-qubit parity operators can be measured directly. We show that the SSC has error threshold approximately 0.97% in this setting.07/2012; -
Article: Practical learning method for multi-scale entangled states
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ABSTRACT: We describe a method for reconstructing multi-scale entangled states from a small number of efficiently-implementable measurements and fast post-processing. The method only requires single particle measurements and the total number of measurements is polynomial in the number of particles. Data post-processing for state reconstruction uses standard tools, namely matrix diagonalisation and conjugate gradient method, and scales polynomially with the number of particles. Our method prevents the build-up of errors from both numerical and experimental imperfections.04/2012; -
Article: Practical characterization of quantum devices without tomography.
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ABSTRACT: Quantum tomography is the main method used to assess the quality of quantum information processing devices. However, the amount of resources needed for quantum tomography is exponential in the device size. Part of the problem is that tomography generates much more information than is usually sought. Taking a more targeted approach, we develop schemes that enable (i) estimating the fidelity of an experiment to a theoretical ideal description, (ii) learning which description within a reduced subset best matches the experimental data. Both these approaches yield a significant reduction in resources compared to tomography. In particular, we demonstrate that fidelity can be estimated from a number of simple experiments that is independent of the system size, removing an important roadblock for the experimental study of larger quantum information processing units.Physical Review Letters 11/2011; 107(21):210404. · 7.37 Impact Factor -
Article: Practical Characterization of Quantum Devices without Tomography
Physical Review Letters 11/2011; 107(21):210404. · 7.37 Impact Factor -
Article: Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space.
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ABSTRACT: We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space. This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time. We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a well-known counting argument. This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model.Physical Review Letters 04/2011; 106(17):170501. · 7.37 Impact Factor -
Article: Universal topological phase of 2D stabilizer codes
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ABSTRACT: Two topological phases are equivalent if they are connected by a local unitary transformation. In this sense, classifying topological phases amounts to classifying long-range entanglement patterns. We show that all 2D topological stabilizer codes are equivalent to several copies of one universal phase: Kitaev's topological code. Error correction benefits from the corresponding local mappings.03/2011; -
Article: Markov entropy decomposition: a variational dual for quantum belief propagation.
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ABSTRACT: We present a lower bound for the free energy of a quantum many-body system at finite temperature. This lower bound is expressed as a convex optimization problem with linear constraints, and is derived using strong subadditivity of von Neumann entropy and a relaxation of the consistency condition of local density operators. The dual to this minimization problem leads to a set of quantum belief propagation equations, thus providing a firm theoretical foundation to that approach. The minimization problem is numerically tractable, and we find good agreement with quantum Monte Carlo calculations for spin-1/2 Heisenberg antiferromagnet in two dimensions. This lower bound complements other variational upper bounds. We discuss applications to Hamiltonian complexity theory and give a generalization of the structure theorem of [P. Hayden et al., Commun. Math. Phys. 246, 359 (2004).] to trees in an appendix.Physical Review Letters 02/2011; 106(8):080403. · 7.37 Impact Factor -
Article: Efficient quantum state tomography.
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ABSTRACT: Quantum state tomography--deducing quantum states from measured data--is the gold standard for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because the number of measurements and the amount of computation required to process them grows exponentially in the system size. Here, we present two tomography schemes that scale much more favourably than direct tomography with system size. One of them requires unitary operations on a constant number of subsystems, whereas the other requires only local measurements together with more elaborate post-processing. Both rely only on a linear number of experimental operations and post-processing that is polynomial in the system size. These schemes can be applied to a wide range of quantum states, in particular those that are well approximated by matrix product states. The accuracy of the reconstructed states can be rigorously certified without any a priori assumptions.Nature Communications 12/2010; 1:149. · 7.40 Impact Factor -
Article: A renormalization group decoding algorithm for topological quantum codes
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ABSTRACT: Topological quantum error-correcting codes are defined by geometrically local checks on a two-dimensional lattice of quantum bits (qubits), making them particularly well suited for fault-tolerant quantum information processing. Here, we present a decoding algorithm for topological codes that is faster than previously known algorithms and applies to a wider class of topological codes. Our algorithm makes use of two methods inspired from statistical physics: renormalization groups and mean-field approximations. First, the topological code is approximated by a concatenated block code that can be efficiently decoded. To improve this approximation, additional consistency conditions are imposed between the blocks, and are solved by a belief propagation algorithm.06/2010; -
Article: Lieb-Robinson bound and locality for general markovian quantum dynamics.
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ABSTRACT: The Lieb-Robinson bound shows the existence of a maximum speed of signal propagation in discrete quantum mechanical systems with local interactions. This generalizes the concept of relativistic causality beyond field theory, and provides a powerful tool in theoretical condensed matter physics and quantum information science. Here, we extend the scope of this seminal result by considering general markovian quantum evolution, where we prove that an equivalent bound holds. In addition, we use the generalized bound to demonstrate that correlations in the stationary state of a Markov process decay on a length scale set by the Lieb-Robinson velocity and the system's relaxation time.Physical Review Letters 05/2010; 104(19):190401. · 7.37 Impact Factor -
Article: Efficient Direct Tomography for Matrix Product States
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ABSTRACT: In this note, we describe a method for reconstructing matrix product states from a small number of efficiently-implementable measurements. Our method is exponentially faster than standard tomography, and it can also be used to certify that the unknown state is an MPS. The basic idea is to use local unitary operations to measure in the Schmidt basis, giving direct access to the MPS representation. This compares favorably with recently and independently proposed methods that recover the MPS tensors by performing a variational minimization, which is computationally intractable in certain cases. Our method also has the advantage of recovering any MPS, while other approaches were limited to special classes of states that exclude important examples such as GHZ and W states.02/2010; -
Article: Fast decoders for topological quantum codes.
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ABSTRACT: We present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects. Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size l, our algorithm runs in time logl compared to l{6} needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold.Physical Review Letters 02/2010; 104(5):050504. · 7.37 Impact Factor -
Article: Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer.
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ABSTRACT: We present a quantum algorithm to prepare the thermal Gibbs state of interacting quantum systems. This algorithm sets a universal upper bound D(alpha) on the thermalization time of a quantum system, where D is the system's Hilbert space dimension and alpha < or = 1/2 is proportional to the Helmholtz free energy density. We also derive an algorithm to evaluate the partition function of a quantum system in a time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared.Physical Review Letters 11/2009; 103(22):220502. · 7.37 Impact Factor -
Article: Preparing ground States of quantum many-body systems on a quantum computer.
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ABSTRACT: Preparing the ground state of a system of interacting classical particles is an NP-hard problem. Thus, there is in general no better algorithm to solve this problem than exhaustively going through all N configurations of the system to determine the one with lowest energy, requiring a running time proportional to N. A quantum computer, if it could be built, could solve this problem in time sqrt[N]. Here, we present a powerful extension of this result to the case of interacting quantum particles, demonstrating that a quantum computer can prepare the ground state of a quantum system as efficiently as it does for classical systems.Physical Review Letters 05/2009; 102(13):130503. · 7.37 Impact Factor
Top co-authors
Top Journals
Institutions
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2009–2013
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Université du Québec
Québec, Quebec, Canada
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2009–2011
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Université de Sherbrooke
- Department of Physics
Sherbrooke, Quebec, Canada
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