  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 phase-invariant
operator is one that is unchanged under the relative phase-shift $\ket{\1}
\mapsto e^{i \theta}\ket{\1}$, for any $\theta$, of all of the $n$ qubits. We
show that phase-invariant unitary operators can be implemented exactly with no
copies and that phase-invariant 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 non-phase-invariant 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
phase-shifted 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 quantum-public-key cryptography, however the techniques are
general. We apply our results to quantum-public-key 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).
