151 research outputs found
Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates
In this paper, we show the equivalence of the set of unitaries computable by
the circuits over the Clifford and T library and the set of unitaries over the
ring , in the single-qubit case. We report an
efficient synthesis algorithm, with an exact optimality guarantee on the number
of Hadamard and T gates used. We conjecture that the equivalence of the sets of
unitaries implementable by circuits over the Clifford and T library and
unitaries over the ring holds in the
-qubit case.Comment: 23 pages, 3 figures, added the proof of T-optimality of the circuits
synthesized by Algorithm
Lower bounds on the non-Clifford resources for quantum computations
We establish lower-bounds on the number of resource states, also known as
magic states, needed to perform various quantum computing tasks, treating
stabilizer operations as free. Our bounds apply to adaptive computations using
measurements and an arbitrary number of stabilizer ancillas. We consider (1)
resource state conversion, (2) single-qubit unitary synthesis, and (3)
computational tasks.
To prove our resource conversion bounds we introduce two new monotones, the
stabilizer nullity and the dyadic monotone, and make use of the already-known
stabilizer extent. We consider conversions that borrow resource states, known
as catalyst states, and return them at the end of the algorithm. We show that
catalysis is necessary for many conversions and introduce new catalytic
conversions, some of which are close to optimal.
By finding a canonical form for post-selected stabilizer computations, we
show that approximating a single-qubit unitary to within diamond-norm precision
requires at least
-states on average. This is the first lower bound that applies to synthesis
protocols using fall-back, mixing techniques, and where the number of ancillas
used can depend on .
Up to multiplicative factors, we optimally lower bound the number of or
states needed to implement the ubiquitous modular adder and
multiply-controlled- operations. When the probability of Pauli measurement
outcomes is 1/2, some of our bounds become tight to within a small additive
constant.Comment: 62 page
- β¦