206 research outputs found
Universal Quantum Computation with ideal Clifford gates and noisy ancillas
We consider a model of quantum computation in which the set of elementary
operations is limited to Clifford unitaries, the creation of the state ,
and qubit measurement in the computational basis. In addition, we allow the
creation of a one-qubit ancilla in a mixed state , which should be
regarded as a parameter of the model. Our goal is to determine for which
universal quantum computation (UQC) can be efficiently simulated. To answer
this question, we construct purification protocols that consume several copies
of and produce a single output qubit with higher polarization. The
protocols allow one to increase the polarization only along certain ``magic''
directions. If the polarization of along a magic direction exceeds a
threshold value (about 65%), the purification asymptotically yields a pure
state, which we call a magic state. We show that the Clifford group operations
combined with magic states preparation are sufficient for UQC. The connection
of our results with the Gottesman-Knill theorem is discussed.Comment: 15 pages, 4 figures, revtex
Improved magic states distillation for quantum universality
Given stabilizer operations and the ability to repeatedly prepare a
single-qubit mixed state rho, can we do universal quantum computation? As
motivation for this question, "magic state" distillation procedures can reduce
the general fault-tolerance problem to that of performing fault-tolerant
stabilizer circuits.
We improve the procedures of Bravyi and Kitaev in the Hadamard "magic"
direction of the Bloch sphere to achieve a sharp threshold between those rho
allowing universal quantum computation, and those for which any calculation can
be efficiently classically simulated. As a corollary, the ability to repeatedly
prepare any pure state which is not a stabilizer state (e.g., any single-qubit
pure state which is not a Pauli eigenstate), together with stabilizer
operations, gives quantum universality. It remains open whether there is also a
tight separation in the so-called T direction.Comment: 6 pages, 5 figure
Magic state distillation with low overhead
We propose a new family of error detecting stabilizer codes with an encoding
rate 1/3 that permit a transversal implementation of the pi/8-rotation on
all logical qubits. The new codes are used to construct protocols for
distilling high-quality `magic' states by Clifford group gates and Pauli
measurements. The distillation overhead has a poly-logarithmic scaling as a
function of the output accuracy, where the degree of the polynomial is
. To construct the desired family of codes, we introduce
the notion of a triorthogonal matrix --- a binary matrix in which any pair and
any triple of rows have even overlap. Any triorthogonal matrix gives rise to a
stabilizer code with a transversal -gate on all logical qubits, possibly
augmented by Clifford gates. A powerful numerical method for generating
triorthogonal matrices is proposed. Our techniques lead to a two-fold overhead
reduction for distilling magic states with output accuracy compared
with the best previously known protocol.Comment: 11 pages, 3 figure
Small Codes for Magic State Distillation
Magic state distillation is a critical component in leading proposals for
fault-tolerant quantum computation. Relatively little is known, however, about
how to construct a magic state distillation routine or, more specifically,
which stabilizer codes are suitable for the task. While transversality of a
non-Clifford gate within a code often leads to efficient distillation routines,
it appears to not be a necessary condition. Here we have examined a number of
small stabilizer codes and highlight a handful of which displaying interesting,
albeit inefficient, distillation behaviour. Many of these distill noisy states
right up to the boundary of the known undististillable region, while some
distill toward non-stabilizer states that have not previously been considered.Comment: Some additional comments and clarifications. Close to published
version. Contribution to the Topical Issue "Quantum Information Processing
and Communication". 6 pages, 6 figure
Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count
We present an infinite family of protocols to distill magic states for
-gates that has a low space overhead and uses an asymptotic number of input
magic states to achieve a given target error that is conjectured to be optimal.
The space overhead, defined as the ratio between the physical qubits to the
number of output magic states, is asymptotically constant, while both the
number of input magic states used per output state and the -gate depth of
the circuit scale linearly in the logarithm of the target error (up to
). Unlike other distillation protocols, this protocol
achieves this performance without concatenation and the input magic states are
injected at various steps in the circuit rather than all at the start of the
circuit. The protocol can be modified to distill magic states for other gates
at the third level of the Clifford hierarchy, with the same asymptotic
performance. The protocol relies on the construction of weakly self-dual CSS
codes with many logical qubits and large distance, allowing us to implement
control-SWAPs on multiple qubits. We call this code the "inner code". The
control-SWAPs are then used to measure properties of the magic state and detect
errors, using another code that we call the "outer code". Alternatively, we use
weakly-self dual CSS codes which implement controlled Hadamards for the inner
code, reducing circuit depth. We present several specific small examples of
this protocol.Comment: 39 pages, (v2) renamed "odd" and "even" weakly self-dual CSS codes of
(v1) to "normal" and "hyperbolic" codes, respectively. (v3) published in
Quantu
- …