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Quantum Information and Computation

Exact universality from any entangling gate without inverses

Aram W. Harrow

Published Sep 1, 2009• 9 citations
Publisher

Abstract

This note proves that arbitrary local gates together with any entangling bipartite gate V are universal. Previously this was known only when access to both V and V † was given, or when approximate universality was demanded. A common situation in quantum computing is that we can apply only a limited set S ⊂ Ud of unitary gates to some d-dimensional system. The first question we want to ask in this situation is whether gates from S can (approximately) generate any gate in PUd = Ud/U1 (the set of all d × d unitary matrices up to an overall phase). When this is possible, we say that S is (approximately) universal. See [1, 3, 4, 7] for original work on this subject, or Sect 4.5 of [9] or Chapter 8 of [8] for reviews. Formally, S is universal (for PUd) if, for all W ∈ PUd, there exists U1,...,Uk ∈ S such that W = UkUk−1 · · · U2U1, whereas U is approximately universal (for PUd) if, for all W ∈ PUd and all ǫ> 0, there exists U1,...,Uk ∈ S such that d(W,UkUk−1 · · · U2U1) < ǫ. (1) Here d(·, ·) can be any metric, but for concreteness we will take it to be the PUd analogue of operator distance: 〈ψ|U d(U,V): = 1 − inf

DOI
10.5555/2011804.2011808

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This note proves that arbitrary local gates together with any entangling bipartite gate V are universal. Previously this was known only when access to both V and V † was given, or when approximate universality was demanded. A common situation in quantum computing is that we can apply only a limited set S ⊂ Ud of unitary gates to some d-dimensional system. The first question we want to ask in this situation is whether gates from S can (approximately) generate any gate in PUd = Ud/U1 (the set of all d × d unitary matrices up to an overall phase). When this is possible, we say that S is (approximately) universal. See [1, 3, 4, 7] for original work on this subject, or Sect 4.5 of [9] or Chapter 8 of [8] for reviews. Formally, S is universal (for PUd) if, for all W ∈ PUd, there exists U1,...,Uk ∈ S such that W = UkUk−1 · · · U2U1, whereas U is approximately universal (for PUd) if, for all W ∈ PUd and all ǫ> 0, there exists U1,...,Uk ∈ S such that d(W,UkUk−1 · · · U2U1) < ǫ. (1) Here d(·, ·) can be any metric, but for concreteness we will take it to be the PUd analogue of operator distance: 〈ψ|U d(U,V): = 1 − inf
A well-known theorem states that the Clifford+T gate set is universal for quantum computing. The theorem combines approximation using a distance measure on unitary matrices with equality up to global phase. Previous proofs combine these two notions for specific distance measures, but do not identify the properties that govern how they work together. Which properties are sufficient to state and prove the theorem? We answer this question by defining projective distance measures using four axioms. We prove that the universality theorem holds for every distance measure satisfying these axioms. Thus, our formulation of the theorem is independent of any particular choice of distance measure. We show that the Hilbert-Schmidt distance is already a projective distance measure. We also develop a general construction that transforms a large class of distance measures into projective distance measures and apply it to obtain projective versions of the operator-norm distance, the Frobenius distance, and the trace distance. Finally, we formalize the proof of the universality theorem in Lean.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers.        The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected.       This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate.       Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
We address the problem of universality in simulation of evolution of quantum system and in theory of quantum computations related with the possibility of expression or approximation of arbitrary unitary transformation by composition of specific unitary transformations (quantum gates) from given set. In an earlier paper application of Clifford algebras to constructions of universal sets of binary quantum gates Uk∈U(2n) was shown. For application of a similar approach to nonbinary quantum gates Uk∈U(ln), in present work we used rational noncommutative torus T1/l2n. A set of universal nonbinary two-gates is presented here as one example.
Quantum computation is of current ubiquitous interest in physics, computer science, and the public interest. In the not-so-distant future, quantum computers will be relatively common pieces of research equipment. Eventually, one can expect an actively quantum computer to be a common feature of life. In this work, I study the approximation efficiency of several common universal quantum gate sets at short sequence lengths using an implementation of the Solovay-Kitaev algorithm. I begin by developing from almost nothing the relevant formal mathematics to rigorously describe what one means by the terms universal gate set and covering efficiency. I then describe some interesting results on the asymptotic covering properties of certain classes of universal gate sets and discuss the theorem which the Solovay-Kitaev algorithm is based on. Moving from mathematical introduction to experimental method, I then describe how sets will be compared. I use the commonly studied sets H+T, Pauli+V, V, and Clifford+T to determine which is the most efficient at approximating randomly generated unitaries. By doing so, we get an understanding of how well each set would perform in the context of a general quantum computer processor. This was accomplished by using the same implementation of the Solovay-Kitaev algorithm throughout, with roughly equal-sized preprocessed libraries formed from each gate set, over approximations for 10,000 randomly generated unitary matrices at algorithm depth n=5. Ultimately, the Pauli+V and V sets were the most efficient and had similar performance qualities. On average the Pauli+V set produced approximations of length 15,491 and accuracy 0.0002686. The V basis produced approximations of average sequence length 16,403 and accuracy 0.0001465. This performance is about equal given this particular implementation of the Solovay-Kitaev algorithm. We conclude that this result is somewhat surprising as the general behavior and efficiency of these particular choices of gate set are expected to be similar. It is possible though that the asymptotic efficiencies of these gate sets vary by a relatively wide margin and this has effected the experiment. It is also possible that some aspect of a naive implementation of the Solovay-Kitaev algorithm resulted in the Hadamard gate based sets performing more poorly than the V basis sets overall. Due to constraints on computational power, this result could also be limited to this particular accuracy regime and could even out as tolerance 𝜀 is taken to be arbitrarily small. Further possibilities of this result as well as further work are then discussed.