Universal quantum computation in a hidden basis

• Authors: Lawrence M. Ioannou;Michele Mosca
• Venue: Quantum Information & Computation
• Year: 2010

Let |0〉 and |1〉 be two states that are promised to come from known subsets of orthogonalsubspaces, but are otherwise unknown. Our paper probes the question of what can beachieved with respect to the basis {|0〉, |1〉}⊗n of n logical qubits, given only a few copiesof the unknown states |0〉 and I1〉. A phase-invariant operator is one that is unchangedunder the relative phase-shift |1〉 ↔ eiθ |1〉, for any θ, of all of the n qubits. We showthat phase-invariant unitary operators can be implemented exactly with no copies andthat phase-invariant states can be prepared exactly with at most n copies each of |0〉and |1〉; we give an explicit algorithm for state preparation that is efficient for someclasses of states (e.g. symmetric states). We conjecture that certain non-phase-invariantoperations are impossible to perform accurately without many copies. Motivated byoptical implementations of quantum computers, we define "quantum computation ina hidden basis" to mean executing a quantum algorithm with respect to the phase-shiftedhidden basis {|0〉, eiθ |1θ}, for some potentially unknown θ; we give an efficientapproximation algorithm for this task, for which we introduce an analogue of a coherentstate of light, which serves as a bounded quantum phase reference frame encoding θ. Ourmotivation was quantum-public-key cryptography, however the techniques are general.We apply our results to quantum-public-key authentication protocols, by showing that anatural class of digital signature schemes for classical messages is insecure. We also givea protocol for identification that uses many of the ideas discussed and whose securityrelates to our conjecture (but we do not know if it is secure).