All Pure Fermionic Non-Gaussian States Are Magic States for Matchgate Computations

Hebenstreit, Martin; Jozsa, Richard; Kraus, Barbara; Strelchuk, Sergii; Yoganathan, Mithuna

doi:10.1103/physrevlett.123.080503

Cited by 26 publications

(47 citation statements)

References 38 publications

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“…Regarding the second setting, let us recall [23] that any pure fermionic non-Gaussian state (cf. Sec.…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…Furthermore (cf. [23]) the state | + 13 | + 24 of four qubits consecutively labeled as 1234 (and | + being the standard Bell state) is a magic state. Here we show that MG circuits with arbitrary entangled 2-qubit inputs on neighboring lines, adaptive measurements in the computational basis, and final measurements in any product basis remain efficiently weakly simulable (Theorem 7), generalizing the simulability result of [20] [plus answering the open question (ii) therein].…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…The strikingly different classical simulation complexity properties of Clifford circuits vs MG circuits, particularly in a scenario of product state inputs and adaptive measurements, are also reflected in the notion of magic states as it applies to these two classes of circuits [9,23]. In essence, magic state inputs together with the availability of adaptive measurements provide a means for elevating the computational power of a class of circuits, here Clifford circuits or MG circuits, to universal quantum computing power.…”

Section: Introductionmentioning

confidence: 99%

“…In the Clifford case the magic states can be single-qubit states, which can be chosen to deterministically implement the T gate [9]. In contrast, for MG circuits the simplest pure magic state input is a 4-qubit entangled state [23], while arbitrary 1-qubit state inputs (i.e., product states) with adaptive measurements remain all classically efficiently simulable.…”

Section: Introductionmentioning

confidence: 99%

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Computational power of matchgates with supplementary resources

et al. 2020

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We study the classical simulation complexity, in both the weak and strong senses, of matchgate (MG) computations supplemented with all combinations of settings involving inclusion of intermediate adaptive or nonadaptive computational basis measurements, product state or magic and general entangled state inputs, and single-or multiple-line outputs. We find a striking parallel to known results for Clifford circuits, after some rebranding of resources. We also give bounds on the amount of classical simulation effort required in the case of limited access to intermediate measurements and entangled inputs. In further settings we show that adaptive MG circuits remain classically efficiently simulable if arbitrary two-qubit entangled input states on consecutive lines are allowed, but become quantum universal for three or more lines. And if adaptive measurements in noncomputational bases are allowed, even with just computational basis inputs, we get quantum universal power again.

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“…Regarding the second setting, let us recall [23] that any pure fermionic non-Gaussian state (cf. Sec.…”

Section: Summary Of Resultsmentioning

confidence: 99%

Section: Summary Of Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computational power of matchgates with supplementary resources

et al. 2020

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“…for a qudit (that is, d-dimensional quantum state). Thus, we can easily observe that the above construction (i.e., ε-FPQC) has very similar structure to the n-qubit secure protocol for quantum sequential transmission [29], magic-state construction [30], and an operator system in mathematics for the error correction schemes in qubit levels [31].…”

Section: Introductionmentioning

confidence: 91%

Approximate Private Quantum Channels on Fermionic Gaussian Systems

Jeong¹

2021

JQIS

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The private quantum channel (PQC) maps any quantum state to the maximally mixed state for the discrete as well as the bosonic Gaussian quantum systems, and it has fundamental meaning on the quantum cryptographic tasks and the quantum channel capacity problems. In this paper, we primally introduce a notion of approximate private quantum channel (ε-PQC) on fermionic Gaussian systems (i.e., ε-FPQC), and construct its explicit form of the fermionic (Gaussian) private quantum channel. First of all, we suggest a general structure for ε-FPQC on the fermionic Gaussian systems with respect to the Schatten p-norm class, and then we give an explicit proof of the statement in the trace norm case. In addition, we study that the cardinality of a set of fermionic unitary operators agrees on the ε-FPQC condition in the trace norm case. This result may give birth to intuition on the construction of emerging fermionic Gaussian quantum communication or computing systems.

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Free Fermions Behind the Disguise

Elman

Chapman

Flammia

2021

Commun. Math. Phys.

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An invaluable method for probing the physics of a quantum many-body spin system is a mapping to noninteracting effective fermions. We find such mappings using only the frustration graph G of a Hamiltonian H, i.e., the network of anticommutation relations between the Pauli terms in H in a given basis. Specifically, when G is (even-hole, claw)-free, we construct an explicit free-fermion solution for H using only this structure of G, even when no Jordan-Wigner transformation exists. The solution method is generic in that it applies for any values of the couplings. This mapping generalizes both the classic Lieb-Schultz-Mattis solution of the XY model and an exact solution of a spin chain recently given by Fendley, dubbed "free fermions in disguise." Like Fendley's original example, the free-fermion operators that solve the model are generally highly nonlinear and nonlocal, but can nonetheless be found explicitly using a transfer operator defined in terms of the independent sets of G. The associated single-particle energies are calculated using the roots of the independence polynomial of G, which are guaranteed to be real by a result of Chudnovsky and Seymour. Furthermore, recognizing (even-hole, claw)-free graphs can be done in polynomial time, so recognizing when a spin model is solvable in this way is efficient. In a crucial step to proving our result, we additionally prove that there exists a hierarchy of commuting conserved charges for models whose frustration graphs are claw-free only, and hence these models are integrable. Finally, we give several example families of solvable and integrable models for which no Jordan-Wigner solution exists, and we give a detailed analysis of such a spin chain having 4-body couplings using this method.

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All Pure Fermionic Non-Gaussian States Are Magic States for Matchgate Computations

Cited by 26 publications

References 38 publications

Computational power of matchgates with supplementary resources

Computational power of matchgates with supplementary resources

Approximate Private Quantum Channels on Fermionic Gaussian Systems

Free Fermions Behind the Disguise

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