Universal Quantum Gates

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. 1. Statements of main results We determine which 2-qudit gates V have the property that all 1-qudit gates together with V form a universal collection, in either the approximate sense or the exact sense. Here d is an arbitrary integer ≥ 2. Our results are new for the case of qubits, i.e., d = 2 (which for many is the case of primary interest). We treat the case d > 2 as well because it is of independent interest and requires no additional work. Since Deutsch [3] found a universal gate (on 3 qubits), universal gates for qubits have been extensively studied. We mention in particular the papers [1], [2] [4], [5] and [6] which will be further discussed in §2. First we set up some notations. A qudit is a (normalized) state in the Hilbert space C. An n-qudit is a state in the tensor product Hilbert space H = (C) = C ⊗ · · · ⊗ C. The computational basis of H is the orthonormal basis given by the d classical n-qudits |i1i2 · · · in〉 = |i1〉 ⊗ |i2〉 ⊗ · · · ⊗ |in〉 (1.1) where 0 ≤ ij ≤ d− 1. The general state in H is a superposition