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Criger, Ben; Terhal, Barbara M.

doi:10.26421/qic16.15-16

Cited by 22 publications

(24 citation statements)

References 35 publications

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“…Since the qubit stabiliser state vectors in Hilbert space form a complex projective 3-design [25,40,41], we get the following corollary: Corollary 1. The group of stabiliser symmetries Aut(SP n ) is given by the adjoint representation of the extended Clifford group EC n .…”

Section: Symmetries Of 3-designsmentioning

confidence: 99%

Robustness of Magic and Symmetries of the Stabiliser Polytope

Heinrich

Gross

2019

Quantum

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We give a new algorithm for computing the robustness of magic-a measure of the utility of quantum states as a computational resource. Our work is motivated by the magic state model of fault-tolerant quantum computation. In this model, all unitaries belong to the Clifford group. Non-Clifford operations are effected by injecting nonstabiliser states, which are referred to as magic states in this context. The robustness of magic measures the complexity of simulating such a circuit using a classical Monte Carlo algorithm. It is closely related to the degree negativity that slows down Monte Carlo simulations through the infamous sign problem. Surprisingly, the robustness of magic is submultiplicative. This implies that the classical simulation overhead scales subexponentially with the number of injected magic states-better than a naive analysis would suggest. However, determining the robustness of n copies of a magic state is difficult, as its definition involves a convex optimisation problem in a 4 n -dimensional space. In this paper, we make use of inherent symmetries to reduce the problem to n dimensions. The total run-time of our algorithm, while still exponential in n, is super-polynomially faster than previously published methods. We provide a computer implementation and give the robustness of up to 10 copies of the most commonly used magic states. Guided by the exact results, we find a finite hierarchy of approximate solutions where each level can be evaluated in polynomial time and yields rigorous upper bounds to the robustness. Technically, we use symmetries of the stabiliser polytope to connect the robustness of magic to the geometry of a low-dimensional convex polytope generated by certain signed quantum weight enumerators. As a by-product, we characterised the automorphism group of the stabiliser polytope, and, more generally, of projections onto complex projective 3-designs. R reg (ρ) := lim n→∞ R(ρ ⊗n ) 1/n .Unfortunately, computing R(ρ ⊗n ) seems to be a difficult task. For ρ being a singlequbit state, the tensor power ρ ⊗n lives in an 4 n -dimensional space, and the sum over the s i in the definition (2) of the RoM has to range over the 2 O(n 2 ) stabiliser states defined for n-qubit systems. Any direct implementation of the optimisation problem (2) will thus quickly became computationally intractable-and, indeed, Howard and Campbell [23] could carry it out only up to n = 5.The starting point of this work is the observation that there is a large symmetry group shared by ρ ⊗n and the stabiliser polytope. Thus, we formulate the optimisation

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Section: Symmetries Of 3-designsmentioning

confidence: 99%

Robustness of Magic and Symmetries of the Stabiliser Polytope

Heinrich

Gross

2019

Quantum

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“…These operator products can also be used to expand the master equation in actual calculations [27,31]. In recent years the permutation symmetric method has been independently rediscovered by different groups using different approaches [22,26,32,33,44,45].…”

Section: Permutation Symmetric Methodsmentioning

confidence: 99%

“…The permutation symmetry of the master equation equation (2) confines the dynamics of the density matrix onto the subspace of symmetrized Liouville space states [32,33,35,46]:…”

Section: Appendix a Details To The Permutation Symmetric Methodsmentioning

confidence: 99%

“…The formal permutation symmetry of the master equation itself allows, under some simple hypothesis on the initial state, to reduce the complexity from an exponential scaling in N to a polynomial scaling ∼N 3 , even without total spin conservation [11,22,[30][31][32][33][34]. This makes exact calculations for moderate emitter numbers feasible and removes constraints imposed by assumptions and approximations.…”

Section: Introductionmentioning

confidence: 99%

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Superradiant to subradiant phase transition in the open system Dicke model: dark state cascades

et al. 2018

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Collectivity in ensembles of atoms gives rise to effects like super-and subradiance. While superradiance is well studied and experimentally accessible, subradiance remains elusive since it is difficult to track experimentally as well as theoretically. Here we present a new type of phase transition in the resonantly driven, open Dicke model that leads to a deterministic generation of subradiant states. At the transition the system switches from a predominantly superradiant to a predominantly subradiant state. Clear experimental signatures for the effect are presented and entanglement properties are discussed. Letting the system relax into the ground state generates a cascade of dark Dicke states, with dark state populations up to unity. Furthermore we introduce a collectivity measure that allows to quantify collective behavior.

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“…2 For bosonic codes, concatenation means that each physical qubit of a code is replaced by another logical qubit. For instance, a 4.8.8 bosonic color code can be obtained by concatenating a bosonic surface code with a [ [4,2,2]] code [36]. This means that two bosonic surface codes are stacked on top of each other, and pairs of stacked qubits are replaced by [[4, 2, 2]] codes, as shown in Fig.…”

Section: Majorana Color Codesmentioning

confidence: 99%

Quantum computing with Majorana fermion codes

Litinski

Oppen

2018

Phys. Rev. B

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We establish a unified framework for Majorana-based fault-tolerant quantum computation with Majorana surface codes and Majorana color codes. All logical Clifford gates are implemented with zero time overhead. This is done by introducing a protocol for Pauli product measurements with tetrons and hexons which only requires local 4-Majorana parity measurements. An analogous protocol is used in the fault-tolerant setting, where tetrons and hexons are replaced by Majorana surface code patches, and parity measurements are replaced by lattice surgery, still only requiring local few-Majorana parity measurements. To this end, we discuss twist defects in Majorana fermion surface codes and adapt the technique of twist-based lattice surgery to fermionic codes. Moreover, we propose a new family of codes that we refer to as Majorana color codes, which are obtained by concatenating Majorana surface codes with small Majorana fermion codes. Majorana surface and color codes can be used to decrease the space overhead and stabilizer weight compared to their bosonic counterparts. arXiv:1801.08143v2 [cond-mat.mes-hall]

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Cited by 22 publications

References 35 publications

Robustness of Magic and Symmetries of the Stabiliser Polytope

Robustness of Magic and Symmetries of the Stabiliser Polytope

Superradiant to subradiant phase transition in the open system Dicke model: dark state cascades

Quantum computing with Majorana fermion codes

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