Stabilizer rank and higher-order Fourier analysis

Labib, Farrokh

doi:10.22331/q-2022-02-09-645

Cited by 8 publications

(1 citation statement)

References 19 publications

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“…It is also notoriously difficult to prove that the stabiliser rank of |T⟩ ⊗n grows exponentially in n. It must grow at least superpolynomially unless P = NP [24]. Despite the sophisticated areas of mathematics brought to bear on this problem-including ultra-metric matrices [25], higher-order Fourier theory [26], number theory/algebraic geometry [27], probability theory [28], and Boolean analysis [29]-only a linear lower bound has been achieved.…”

Section: Stabiliser Rankmentioning

confidence: 99%

Bases for optimising stabiliser decompositions of quantum states

de Silva,

Yin,

Strelchuk

2024

Quantum Sci. Technol.

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Stabiliser states play a central role in the theory of quantum computation. For example, they are used to encode computational basis states in the most common quantum error correction schemes. Arbitrary quantum states admit many stabiliser decompositions: ways of being expressed as a superposition of stabiliser states. Understanding the structure of stabiliser decompositions has significant applications in verifying and simulating near-term quantum computers.We introduce and study the vector space of linear dependencies of $n$-qubit stabiliser states. These spaces have canonical bases containing vectors whose size grows exponentially in $n$. We construct elegant bases of linear dependencies of constant size three. Critically, our sparse bases can be computed without first compiling a dictionary of all $n$-qubit stabiliser states. We utilise them to explicitly compute the stabiliser extent of states of more qubits than is feasible with existing techniques. Finally, we delineate future applications to improving theoretical bounds on the stabiliser rank of magic states.

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Section: Stabiliser Rankmentioning

confidence: 99%

Bases for optimising stabiliser decompositions of quantum states

de Silva,

Yin,

Strelchuk

2024

Quantum Sci. Technol.

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Optimality of the Howard-Vala T-gate in stabilizer quantum computation

Feng,

Luo

2024

Phys. Scr.

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In a remarkable work [Phys. Rev. A 86, 022316 (2012)], Howard and Vala introduced a qudit version of the qubit T -gate (i.e., π/8-gate) for any prime dimensional system. This non-Cliﬀord gate is a key ingredient of the paradigm “Cliﬀord+T ”, which are widely employed in the stabilizer formalism of universal and fault-tolerant quantum computation. Considering the applications and signiﬁcance of the T -gate, it is desirable to characterize it from various angles. Here we prove that in any prime dimensional system, the Howard-Vala T -gate is optimal, among all diagonal gates, for generating magic resources from stabilizer states when the magic is quantiﬁed via the L1-norm of characteristic functions (Fourier transforms) of quantum states. The quadratic Gaussian sum in number theory plays a key role in establishing this optimality. This highlights an extreme feature of the Howard-Vala T -gate. We further reveal an intrinsic relation between the Howard-Vala T -gate and the Watson-Campbell-Anwar-Browne T -gate [Phys. Rev. A 92, 022312 (2015)] for any prime dimensional system.dimensional system.

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Group frames via magic states with applications to SIC-POVMs and MUBs

Feng,

Luo

2024

Commun. Theor. Phys.

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We connect magic (non-stabilizer) states, symmetric informationally complete positive operator valued measures (SIC-POVMs), and mutually unbiased bases (MUBs) in the context of group frames, and study their interplay. Magic states are quantum resources in the stabilizer formalism of quantum computation. SIC-POVMs and MUBs are fundamental structures in quantum information theory with many applications in quantum foundations, quantum state tomography, and quantum cryptography, etc. In this work, we study group frames constructed from some prominent magic states, and further investigate their applications. Our method exploits the orbit of discrete Heisenberg–Weyl group acting on an initial fiducial state. We quantify the distance of the group frames from SIC-POVMs and MUBs, respectively. As a simple corollary, we reproduce a complete family of MUBs of any prime dimensional system by introducing the concept of MUB fiducial states, analogous to the well-known SIC-POVM fiducial states. We present an intuitive and direct construction of MUB fiducial states via quantum T-gates, and demonstrate that for the qubit system, there are twelve MUB fiducial states, which coincide with the H-type magic states. We compare MUB fiducial states and SIC-POVM fiducial states from the perspective of magic resource for stabilizer quantum computation. We further pose the challenging issue of identifying all MUB fiducial states in general dimensions.

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Stabilizer rank and higher-order Fourier analysis

Cited by 8 publications

References 19 publications

Bases for optimising stabiliser decompositions of quantum states

Bases for optimising stabiliser decompositions of quantum states

Optimality of the Howard-Vala T-gate in stabilizer quantum computation

Group frames via magic states with applications to SIC-POVMs and MUBs

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