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arXiv:quant-ph/0108062

Universal quantum gates

Jean-Luc Brylinski, Ranee Kathryn Brylinski

Published Aug 13, 2001• 0 citations
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Abstract

In this paper we study universality for quantum gates acting on qudits.Qudits are states in a Hilbert space of dimension d where d is at least two. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit gates together with V produces all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions on V, and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor. We discuss some applications and also relations with work of other authors.

DOI
10.48550/arxiv.quant-ph/0108062

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In this paper we study universality for quantum gates acting on qudits.Qudits are states in a Hilbert space of dimension d where d is at least two. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit gates together with V produces all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions on V, and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor. We discuss some applications and also relations with work of other authors.
In this paper we study universality for quantum gates acting on qudits.Qudits are states in a Hilbert space of dimension d where d can be any integer ≥ 2. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit gates together with V produces all n-qudit gates exactly.We show that (i) and (ii) are equivalent conditions on V , and they hold if and only if V is not a primitive gate.Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor.We discuss some applications and also relations with work of other authors.
Abstract In this paper we study universality for quantum gates acting on qudits. Qudits are states in a Hilbert space of dimension d where d can be any integer ≥ 2. We determine which 2-qudit gates V have the properties: (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit gates together with V produces all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions on V , and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor. We discuss some applications and also relations with work of other authors.
We describe criteria for implementation of quantum computation in qudits. A qudit is a -dimensional system whose Hilbert space is spanned by states \ensuremath{\mid}0⟩, \ensuremath{\mid}1⟩, \dots{}, . An important earlier work [] describes how to exactly simulate an arbitrary unitary on multiple qudits using a parameter family of single qudit and two qudit gates. That technique is based on the spectral decomposition of unitaries. Here we generalize this argument to show that exact universality follows given a discrete set of single qudit Hamiltonians and one two-qudit Hamiltonian. The technique is related to the -matrix decomposition of numerical linear algebra. We consider a generic physical system in which the single qudit Hamiltonians are a small collection of and . A coupling graph results taking nodes 0, \dots{}, and edges are allowed Hamiltonians. One qudit exact universality follows iff this graph is connected, and complete universality results if the two-qudit Hamiltonian is also allowed. We discuss implementation in the eight dimensional ground electronic states of and construct an optimal gate sequence using Raman laser pulses.
Quantum computing with qudits, quantum systems with d>2d > 2d>2 levels, offers a powerful extension beyond qubits, expanding the computational possibilities of quantum systems, allowing the simplification of the implementation of several algorithms and, possibly, providing a foundation for optimised error correction. In this work, we propose a novel formulation of qudit gates that is universally applicable for any number of levels ddd, without restrictions on the dimensionality. By extending the mathematical framework of quantum gates to arbitrary dimensions, we derive explicit gate operations that form a universal set for quantum computation on qudits of any size. We demonstrate the validity of our approach through the implementation of the Quantum Fourier Transform (QFT) for arbitrary ddd, verifying both the correctness and utility of our generalized gates. This novel methodology broadens the design space for quantum algorithms and fault-tolerant architectures, paving the way for advancements in qudit-based quantum computing.
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.
We construct solutions of the Yang–Baxter equation in any dimension d≥2d\ge 2d≥2 by directly generalizing the previously found solutions for d=2d=2d=2 . We equip those solutions with unitarity and entangling properties. Being unitary, they can be turned into 222 -qudit quantum logic gates for qudit-based systems. The entangling property enables each of those solutions, together with all 111 -qudit gates, to form a universal set of quantum logic gates.