Computational power and correlation in a quantum computational tensor network

Fujii, Keisuke; Morimae, Tomoyuki

doi:10.1103/physreva.85.032338

Cited by 20 publications

(25 citation statements)

References 31 publications

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“…Previous work on computational phases has investigated the cluster model in external fields (also at non-zero temperature) [7,8], with competing Ising interactions [9], in three dimensions at non-zero temperature [10], with errorsuppressing interacting cluster terms [11], and under general symmetry preserving perturbations [12,13]. For two-body models, studies have looked at the Haldane phase in one dimension [14], an anisotropic AKLT model on a honeycomb lattice [15] and versions of the Hamiltonians in [6], which can also be universal at non-zero temperature [16,17].…”

Section: Introductionmentioning

confidence: 99%

Graph states as ground states of two-body frustration-free Hamiltonians

Darmawan¹,

Bartlett²

2014

New J. Phys.

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The framework of measurement-based quantum computation (MBQC) allows us to view the ground states of local Hamiltonians as potential resources for universal quantum computation. A central goal in this field is to find models with ground states that are universal for MBQC and that are also natural in the sense that they involve only two-body interactions and have a small local Hilbert space dimension. Graph states are the original resource states for MBQC, and while it is not possible to obtain graph states as exact ground states of two-body Hamiltonians, here we construct two-body frustration-free Hamiltonians that have arbitrarily good approximations of graph states as unique ground states. The construction involves taking a two-body frustration-free model that has a ground state convertible to a graph state with stochastic local operations, then deforming the model such that its ground state is close to a graph state. Each graph state qubit resides in a subspace of a higher dimensional particle. This deformation can be applied to two-body frustration-free Affleck-Kennedy -Lieb-Tasaki (AKLT) models, yielding Hamiltonians that are exactly solvable with exact tensor network expressions for ground states. For the star-lattice AKLT model, the ground state of which is not expected to be a universal resource for MBQC, applying such a deformation appears to enhance the computational power of the ground state, promoting it to a universal resource for MBQC. Transitions in computational power, similar to percolation phase transitions, can be observed when Hamiltonians are deformed in this way. Improving the fidelity of the ground state comes at the cost of a shrinking gap. While analytically proving gap properties for these types of models is difficult in general, we provide a detailed analysis of the deformation of a spin-1 AKLT state to a linear graph state.

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Section: Introductionmentioning

confidence: 99%

Graph states as ground states of two-body frustration-free Hamiltonians

Darmawan¹,

Bartlett²

2014

New J. Phys.

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“…These discoveries have established a new bridge between quantum information and condensed-matter physics. MBQC has also Alice Bob offered a new framework for fault-tolerant quantum computing which achieves a high threshold [34][35][36][37][38][39], The quantumclassical separation in MBQC has enabled us to clarify several relations between the "quantumness" of a resource state and the quantum computational power of MBQC on it [40][41][42][43][44][45], New protocols of secure cloud quantum computing, so-called blind quantum computing, were also developed by using MBQC [46][47][48][49][50][51][52][53][54][55][56][57][58][59][60], In the most general framework of MBQC, we first prepare the resource state a of N = nm qubits, as is shown in Fig. 3.…”

Section: General Mbqcmentioning

confidence: 99%

Minimum heat dissipation in measurement-based quantum computation

Morimae

2015

Phys. Rev. A

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“…Now we can imagine a virtual linear space where A ’s, | R 〉, and | L 〉 live. This virtual linear space is called the correlation space 4 5 6 10 11 12 . Then, the above equation can be interpreted that the “gate operation” is implemented on the “quantum state” | R 〉 in the correlation space.…”

mentioning

confidence: 99%

Not all physical errors can be linear CPTP maps in a correlation space

Morimae

Fujii

2012

Sci Rep

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In the framework of quantum computational tensor network, which is a general framework of measurement-based quantum computation, the resource many-body state is represented in a tensor-network form (or a matrix-product form), and universal quantum computation is performed in a virtual linear space, which is called a correlation space, where tensors live. Since any unitary operation, state preparation, and the projection measurement in the computational basis can be simulated in a correlation space, it is natural to expect that fault-tolerant quantum circuits can also be simulated in a correlation space. However, we point out that not all physical errors on physical qudits appear as linear completely-positive trace-preserving errors in a correlation space. Since the theories of fault-tolerant quantum circuits known so far assume such noises, this means that the simulation of fault-tolerant quantum circuits in a correlation space is not so straightforward for general resource states.

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Computational power and correlation in a quantum computational tensor network

Cited by 20 publications

References 31 publications

Graph states as ground states of two-body frustration-free Hamiltonians

Graph states as ground states of two-body frustration-free Hamiltonians

Minimum heat dissipation in measurement-based quantum computation

Not all physical errors can be linear CPTP maps in a correlation space

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