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arXiv:2103.12303

Universality of swap for qudits: a representation theory approach

James R. van Meter

Published Mar 23, 2021• 2 citations
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Abstract

An open problem of quantum information theory has been to determine under what conditions universal exchange-only computation is possible for qudits encoded on $d$-state systems for $d>2$. This problem can be posed in terms of representation theory by recognizing that each quantum mechanical swap, generated by exchange-interaction, can be identified with a transposition in a symmetric group, each $d$-state system can be identified with the fundamental representation of $SU(d)$, and each encoded qudit can be identified with an irreducible representation of a Lie algebra generated by transpositions. Towards this end we first give a mathematical definition of exchange-only universality in terms of a map from the special unitary algebra on the product of qudits into a representation of a Lie algebra generated by transpositions. We show that this definition is consistent with quantum computing requirements. We then proceed with the task of characterizing universal families of qudits, that is families of encoded qudits admitting exchange-only universality. This endeavor is aided by the fact that the irreducible representations corresponding to qudits are canonically labeled by partitions. In particular we derive necessary and sufficient conditions for universality on one or two such qudits, in terms of simple arithmetic conditions on the associated partitions. We also derive necessary and sufficient conditions for universality on arbitrarily many such qudits, in terms of Littlewood--Richardson coefficients. Among other results, we prove that universal families of multiple qudits are upward closed, that universality is guaranteed for sufficiently many qudits, and that any family that is not universal can be made so by simply adding at most five ancillae. We also obtain results for 2-state systems as a special case.

DOI
10.48550/arxiv.2103.12303

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An open problem of quantum information theory has been to determine under what conditions universal exchange-only computation is possible for qudits encoded on ddd-state systems for d>2d>2d>2. This problem can be posed in terms of representation theory by recognizing that each quantum mechanical swap, generated by exchange-interaction, can be identified with a transposition in a symmetric group, each ddd-state system can be identified with the fundamental representation of SU(d)SU(d)SU(d), and each encoded qudit can be identified with an irreducible representation of a Lie algebra generated by transpositions. Towards this end we first give a mathematical definition of exchange-only universality in terms of a map from the special unitary algebra on the product of qudits into a representation of a Lie algebra generated by transpositions. We show that this definition is consistent with quantum computing requirements. We then proceed with the task of characterizing universal families of qudits, that is families of encoded qudits admitting exchange-only universality. This endeavor is aided by the fact that the irreducible representations corresponding to qudits are canonically labeled by partitions. In particular we derive necessary and sufficient conditions for universality on one or two such qudits, in terms of simple arithmetic conditions on the associated partitions. We also derive necessary and sufficient conditions for universality on arbitrarily many such qudits, in terms of Littlewood--Richardson coefficients. Among other results, we prove that universal families of multiple qudits are upward closed, that universality is guaranteed for sufficiently many qudits, and that any family that is not universal can be made so by simply adding at most five ancillae. We also obtain results for 2-state systems as a special case.
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Abstract A family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all k⩾2k \geqslant 2k⩾2 k ⩾ 2 and all local dimensions d⩾2d \geqslant 2d⩾2 d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions d⩾2d \geqslant 2d⩾2 d ⩾ 2 (spin ⩾1/2\geqslant 1/2⩾1/2 ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for d=3d=3d=3 d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.
Qudits are the natural generalization of qubits, that is, the basic unit of information in quantum computation. This generalization allows for quantum states to be represented with the same cyclic nature present in qubits, while also permitting the basic unit of information to be stored in d-level systems. In this document, we are concerned with studying universality over qudits on arbitrary dimension d. The question of single-qudit universality of a set of operators can be reduced to studying the adjoint representation of the set and determining whether or not its generated group is finite. Thus, we set out to find universal sets of operators in quantum computation over qudits, for any possible value of d. Moreover, we focus on finding “Clifford + T-like” universal sets of operators in arbitrary dimensions, in the hopes of maintaining a strong analog to the qubits case and its most studied universal set. We accomplish this goal by studying how the Clifford group behaves on arbitrary dimensions, and how this behavior relates to the typical Clifford group in prime dimensions. Furthermore, we find that the Clifford group on arbitrary dimension splits as the typical Clifford group on each of the coprime factors of d, which leads to a fundamental difference between the study of universality on prime power dimensions and arbitrary dimensions containing co-prime factors. Then, we focus on studying the structure of universal sets in dimension d, for any d containing at least two co-prime factors. Thus, after studying which T-like gates drive the single-qudit Clifford group to universality on prime power dimensions; that is, guaranteeing the irreducibility of the adjoint representation and an infinite generated group, we combine these two approaches to obtain “Clifford + T-like” universal sets on all possible dimensions.
Controllability and observability of multi-spin systems in architectures of various symmetries of coupling type and topology are investigated. We complement recent work of explicitly determining the respective dynamic system Lie algebras and thereby also precise reachability sets under symmetry constraints. Here the focus is on the converse: under which conditions can the absence of symmetry be taken as an indicator of universality of the hardware architecture? More precisely, the absence of symmetry implies irreducibility and provides a convenient necessary condition for full controllability. Though much easier to assess than the well-established Lie-algebra rank condition, this is not sufficient unless in an n-qubit system with connected coupling topology the candidate dynamic simple Lie algebra can be identified uniquely as the full unitary algebra su(2n). - Here we discuss simple tests confined to solving homogeneous linear equations in order to filter irreducible unitary representations of other candidate algebras of classical type (orthogonal ones and unitary symplectic ones). Finally, we give an outlook under which conditions algebras of exceptional type can also be ruled out.
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