177 research outputs found
Hamiltonian Simulation by Qubitization
We present the problem of approximating the time-evolution operator
to error , where the Hamiltonian is the
projection of a unitary oracle onto the state created by
another unitary oracle. Our algorithm solves this with a query complexity
to both oracles that is optimal
with respect to all parameters in both the asymptotic and non-asymptotic
regime, and also with low overhead, using at most two additional ancilla
qubits. This approach to Hamiltonian simulation subsumes important prior art
considering Hamiltonians which are -sparse or a linear combination of
unitaries, leading to significant improvements in space and gate complexity,
such as a quadratic speed-up for precision simulations. It also motivates
useful new instances, such as where is a density matrix. A key
technical result is `qubitization', which uses the controlled version of these
oracles to embed any in an invariant subspace. A large
class of operator functions of can then be computed with optimal
query complexity, of which is a special case.Comment: 23 pages, 1 figure; v2: updated notation; v3: accepted versio
Deterministic and cascadable conditional phase gate for photonic qubits
Previous analyses of conditional \phi-phase gates for photonic qubits that
treat cross-phase modulation (XPM) in a causal, multimode, quantum field
setting suggest that a large (~\pi rad) nonlinear phase shift is always
accompanied by fidelity-degrading noise [J. H. Shapiro, Phys. Rev. A 73, 062305
(2006); J. Gea-Banacloche, Phys. Rev. A 81, 043823 (2010)]. Using an atomic
V-system to model an XPM medium, we present a conditional phase gate that, for
sufficiently small nonzero \phi, has high fidelity. The gate is made cascadable
by using using a special measurement, principal mode projection, to exploit the
quantum Zeno effect and preclude the accumulation of fidelity-degrading
departures from the principal-mode Hilbert space when both control and target
photons illuminate the gate
Universal fault-tolerant gates on concatenated stabilizer codes
It is an oft-cited fact that no quantum code can support a set of
fault-tolerant logical gates that is both universal and transversal. This no-go
theorem is generally responsible for the interest in alternative universality
constructions including magic state distillation. Widely overlooked, however,
is the possibility of non-transversal, yet still fault-tolerant, gates that
work directly on small quantum codes. Here we demonstrate precisely the
existence of such gates. In particular, we show how the limits of
non-transversality can be overcome by performing rounds of intermediate
error-correction to create logical gates on stabilizer codes that use no
ancillas other than those required for syndrome measurement. Moreover, the
logical gates we construct, the most prominent examples being Toffoli and
controlled-controlled-Z, often complete universal gate sets on their codes. We
detail such universal constructions for the smallest quantum codes, the 5-qubit
and 7-qubit codes, and then proceed to generalize the approach. One remarkable
result of this generalization is that any nondegenerate stabilizer code with a
complete set of fault-tolerant single-qubit Clifford gates has a universal set
of fault-tolerant gates. Another is the interaction of logical qubits across
different stabilizer codes, which, for instance, implies a broadly applicable
method of code switching.Comment: 18 pages + 5 pages appendix, 12 figure
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