Universal quantum computation in a hidden basis
Abstract
Let $\ket{\0}$ and $\ket{\1}$ be two states that are promised to come from known subsets of orthogonal subspaces, but are otherwise unknown. Our paper probes the question of what can be achieved with respect to the basis $\{\ket{\0},\ket{\1}}^{\otimes n}$ of $n$ logical qubits, given only a few copies of the unknown states $\ket{\0}$ and $\ket{\1}$. A phaseinvariant operator is one that is unchanged under the relative phaseshift $\ket{\1} \mapsto e^{i \theta}\ket{\1}$, for any $\theta$, of all of the $n$ qubits. We show that phaseinvariant unitary operators can be implemented exactly with no copies and that phaseinvariant states can be prepared exactly with at most $n$ copies each of $\ket{\0}$ and $\ket{\1}$; we give an explicit algorithm for state preparation that is efficient for some classes of states (e.g. symmetric states). We conjecture that certain nonphaseinvariant operations are impossible to perform accurately without many copies. Motivated by optical implementations of quantum computers, we define ``quantum computation in a hidden basis'' to mean executing a quantum algorithm with respect to the phaseshifted hidden basis $\{\ket{\0}, e^{i\theta}\ket{\1}\}$, for some potentially unknown $\theta$; we give an efficient approximation algorithm for this task, for which we introduce an analogue of a coherent state of light, which serves as a bounded quantum phase reference frame encoding $\theta$. Our motivation was quantumpublickey cryptography, however the techniques are general. We apply our results to quantumpublickey authentication protocols, by showing that a natural class of digital signature schemes for classical messages is insecure. We also give a protocol for identification that uses many of the ideas discussed and whose security relates to our conjecture (but we do not know if it is secure).
 Publication:

arXiv eprints
 Pub Date:
 October 2008
 arXiv:
 arXiv:0810.2780
 Bibcode:
 2008arXiv0810.2780I
 Keywords:

 Quantum Physics
 EPrint:
 final version