[Show abstract][Hide abstract] ABSTRACT: In leading fault-tolerant quantum computing schemes, accurate transformation
are obtained by a two-stage process. In a first stage, a discrete, universal
set of fault-tolerant operations is obtained by error-correcting 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
[Show abstract][Hide abstract] ABSTRACT: Steane's 7-qubit quantum error-correcting code admits a set of fault-tolerant
gates that generate the Clifford group, which in itself is not universal for
quantum computation. The 15-qubit Reed-Muller code also does not admit a
universal fault-tolerant gate set but possesses fault-tolerant T and
control-control-Z 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 fault-tolerantly convert between these two codes,
providing a new method to realize universal fault-tolerant 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 Reed-Muller codes.
[Show abstract][Hide abstract] ABSTRACT: We introduce a new class of circuits for constructing efficiently decodable
error-correction codes, based on a recently discovered contractible tensor
network. We perform an in-depth 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 log-linear decoding complexity using the successive
cancellation decoder. These codes also display improved error-correcting
capability with only a minor impact on decoding complexity. Efficient decoding
is realized using powerful graphical calculus tools developed in the field of
quantum many-body physics. In a companion paper, we generalize our construction
to the quantum setting and describe more in-depth the relation between
classical and quantum error correction and the graphical calculus of tensor
[Show abstract][Hide abstract] ABSTRACT: We establish several relations between quantum error correction (QEC) and
tensor network (TN) methods of quantum many-body physics. We exhibit
correspondences between well-known 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 two-dimensional lattice models that support Ising anyonic
excitations and are coupled to a thermal bath. We propose a phenomenological
model for the resulting short-time dynamics that includes pair-creation,
hopping, braiding, and fusion of anyons. By explicitly constructing topological
quantum error-correcting 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 non-Abelian setting. We perform
large-scale numerical simulations of these two-dimensional 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 NP-complete, and a
similar decoding problem for quantum codes is also known to be NP-complete.
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.
[Show abstract][Hide abstract] ABSTRACT: We study the quantum error correction threshold of Kitaev's toric code over the group Zd subject to a generalized bit-flip noise. This problem requires special decoding techniques, and for this purpose we generalize the renormalization-group method we introduced previously [ G. Duclos-Cianci and D. Poulin Phys. Rev. Lett. 104 050504 (2010) and IEEE Information Theory Workshop, Dublin (2010), p. 1] for Z2 topological codes.
Physical Review A 06/2013; 87(6). · 3.04 Impact Factor
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] ABSTRACT: Studying large many-body 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 finite-temperature 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 lower-bound to the ground state energy
(which compliments rigorous upper-bounds that can be found with more
common, variational techniques).
[Show abstract][Hide abstract] ABSTRACT: The Markov entropy decomposition (MED) is a recently-proposed, cluster-based
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 spin-1/2 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 exact-diagonalization, 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 long-range 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 two-dimensional (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.
New Journal of Physics 07/2012; 14(7):073048. · 4.06 Impact Factor
[Show abstract][Hide abstract] 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.
[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 many-body systems---complementing the area law at finite
temperature---and 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 multi-qubit
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.
Physical Review A 04/2012; 86(3). · 3.04 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a decoding algorithm for quantum convolutional codes that finds
the class of degenerate errors with the largest probability conditioned on a
given error syndrome. The algorithm runs in time linear with the number of
qubits. Previous decoding algorithms for quantum convolutional codes optimized
the probability over individual errors instead of classes of degenerate errors.
Using Monte Carlo simulations, we show that this modification to the decoding
algorithm results in a significantly lower block error rate.
IEEE Transactions on Information Theory 04/2012; · 2.62 Impact Factor
[Show abstract][Hide abstract] 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.
New Journal of Physics 04/2012; 14(8). · 4.06 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 2D topological stabilizer codes have attracted a lot of attention in
recent years for two main reasons. First, they provide exactly solvable
models which exhibit topological order and anyonic excitations. Second,
they naturally lead to quantum stabilizer error-correcting codes having
macroscopic minimum distance. Although these codes are robust at zero
temperature, quasi-particles appear and freely diffuse in the system at
any finite temperature. If this diffusion is unchecked, errors will
occur. Consequently, active error-correction is needed. We want to
propose a cellular automaton that would perform this correction. It
would ``manually'' confine the quasi-particles by simulating articifial
attraction between them and moving them accordingly. We obtained
encouraging preliminary results for error-correction and hope to
generalize them to fault-tolerance.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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.