A non-commuting stabilizer formalism

Ni, Xiaotong; Buerschaper, Oliver; Nest, Maarten Van den

doi:10.1063/1.4920923

Cited by 27 publications

(70 citation statements)

References 36 publications

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“…However, they can be decomposed as a linear superpositions of fundamental anyonic errors (string operators creating pairs of anyons) with a known effect on the Hilbert space. Moreover, the semion code is not a subgroup of the Pauli group since the complex phases entering its definition (see equation (2)) makes impossible to express its generators in terms of tensor products of Pauli matrices [32,33]. Nevertheless, the semion code is still an additive code: the sum of quantum codewords is also a codeword [34].…”

Section: Summary Of Main Resultsmentioning

confidence: 99%

“…We notice that a previous study [33] attempted to construct a quantum error correction code using the DS model as the starting point. The main difference with our work is that they construct a non-commuting quantum correcting code, whereas we have succeeded in constructing an extension that belongs to the stabilizer formalism.…”

Section: Discussionmentioning

confidence: 99%

“…As a consequence of this, the whole error correction procedure of the off-shell DS code is topological. On the contrary, the topological nature of the non-commuting code in [33] is unproven. Both constructions share the feature of using non-Pauli operators to construct the basic string operators of the model.…”

Section: Discussionmentioning

confidence: 99%

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Quantum error correction with the semion code

et al. 2019

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We present a full quantum error correcting procedure with the semion code: an off-shell extension of the double-semion model. We construct open-string operators that recover the quantum memory from arbitrary errors and closed-string operators that implement the basic logical operations for information processing. Physically, the new open-string operators provide a detailed microscopic description of the creation of semions at their end-points. Remarkably, topological properties of the string operators are determined using fundamental properties of the Hamiltonian, namely, the fact that it is composed of commuting local terms squaring to the identity. In all, the semion code is a topological code that, unlike previously studied topological codes, it is of non-CSS type and fits into the stabilizer formalism. This is in sharp contrast with previous attempts yielding non-commutative codes.

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Section: Summary Of Main Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Quantum error correction with the semion code

et al. 2019

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“…39 This class of pure states have been studied in the context of locally maximally entangleable states, [40][41][42] and in relation with mutually unbiased bases. 43 They also reveal typical properties of states that appear in instantaneous quantum polynomial-time (IQP) circuits 44,45 and commuting circuits, 46 which are likely to have stronger computational power than classical computers even though they exploit only diagonal gates and a separable pure initial state. However, it had not been yet clari¯ed whether or not the corresponding phase-random states can be e±ciently generated.…”

Section: Introductionmentioning

confidence: 99%

Diagonal-Unitary 2-Design and Their Implementations by Quantum Circuits

Nakata

Murao

2013

Int. J. Quantum Inform.

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We study e±cient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitary t-designs and present two quantum circuits that implement diagonal-unitary 2-design with the computational basis in N -qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary 2-design after applying the gates on all pairs of qubits. The number of required gates is NðN À 1Þ=2. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact 2-design, but achieves an -approximate 2-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most OðN 2 ðN þ log 1=ÞÞ. We also provide an application of the circuits, a protocol of generating an exact 2-design of random states by combining the circuits with a simple classical procedure requiring OðNÞ random classical bits.

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“…Stabilizer states generalize graph states through additional local Clifford operations [55][56][57], which impose parity constraints and extra phases. XS-stabilizer states [32] combines 3-body correlation factors from hypergraph state and parity constraints from stabilizer states. A parity constraint: (v 1 + v 2 + · · · + v k ) mod 2 = 0 can be realized by a hidden neuron that connects to each of these visible neurons v 1 , v 2 , · · · , v k with weight function W (v, h) = iπvh − (ln 2)/4 [58].…”

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confidence: 99%

Efficient representation of topologically ordered states with restricted Boltzmann machines

Gao

Duan

2019

Phys. Rev. B

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Representation by neural networks, in particular by restricted Boltzmann machines (RBM), has provided a powerful computational tool to solve quantum many-body problems. An important open question is how to characterize which class of quantum states can be efficiently represented with RBMs. Here, we show that RBMs can efficiently represent a wide class of many-body entangled states with rich exotic topological orders. This includes: (1) ground states of double semion and twisted quantum double models with intrinsic topological orders; (2) states of the AKLT model and two-dimensional CZX model with symmetry protected topological orders; (3) states of Haah code model with fracton topological order; (4) (generalized) stabilizer states and hypergraph states that are important for quantum information protocols. One twisted quantum double model state considered here harbors non-abelian anyon excitations. Our result shows that it is possible to study a variety of quantum models with exotic topological orders and rich physics using the RBM computational toolbox.Introduction.-Deep learning has become a powerful tool with wide applications [1,2]. Recently, deep learning methods have attracted considerable attention in quantum physics [3,4], especially for attacking quantum many-body problems. The difficulty of quantum manybody problems mainly originates from the exponential growth of the Hilbert space dimension. To overcome this exponential difficulty, researchers traditionally use tensor network methods [5][6][7] and Quantum Monte Carlo (QMC) simulation [8]. However, QMC methods suffer from the sign problem [9]; Tensor network methods have difficulty to deal with high dimensional systems [10] or systems with massive entanglement [11]. These issues call for mew method.

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A non-commuting stabilizer formalism

Cited by 27 publications

References 36 publications

Quantum error correction with the semion code

Quantum error correction with the semion code

Diagonal-Unitary 2-Design and Their Implementations by Quantum Circuits

Efficient representation of topologically ordered states with restricted Boltzmann machines

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