12 research outputs found
Dequantizing read-once quantum formulas
Quantum formulas, defined by Yao [FOCS '93], are the quantum analogs of
classical formulas, i.e., classical circuits in which all gates have fanout
one. We show that any read-once quantum formula over a gate set that contains
all single-qubit gates is equivalent to a read-once classical formula of the
same size and depth over an analogous classical gate set. For example, any
read-once quantum formula over Toffoli and single-qubit gates is equivalent to
a read-once classical formula over Toffoli and NOT gates. We then show that the
equivalence does not hold if the read-once restriction is removed. To show the
power of quantum formulas without the read-once restriction, we define a new
model of computation called the one-qubit model and show that it can compute
all boolean functions. This model may also be of independent interest.Comment: 14 pages, 8 figures, to appear in proceedings of TQC 201
Resource optimization for fault-tolerant quantum computing
In this thesis we examine a variety of techniques for reducing the resources
required for fault-tolerant quantum computation. First, we show how to simplify
universal encoded computation by using only transversal gates and standard
error correction procedures, circumventing existing no-go theorems. We then
show how to simplify ancilla preparation, reducing the cost of error correction
by more than a factor of four. Using this optimized ancilla preparation, we
develop improved techniques for proving rigorous lower bounds on the noise
threshold.
Additional overhead can be incurred because quantum algorithms must be
translated into sequences of gates that are actually available in the quantum
computer. In particular, arbitrary single-qubit rotations must be decomposed
into a discrete set of fault-tolerant gates. We find that by using a special
class of non-deterministic circuits, the cost of decomposition can be reduced
by as much as a factor of four over state-of-the-art techniques, which
typically use deterministic circuits.
Finally, we examine global optimization of fault-tolerant quantum circuits
under physical connectivity constraints. We adapt techniques from VLSI in order
to minimize time and space usage for computations in the surface code, and we
develop a software prototype to demonstrate the potential savings.Comment: 231 pages, Ph.D. thesis, University of Waterlo
Stabilizer circuit verification
The ubiquity of stabilizer circuits in the design and operation of quantum
computers makes techniques to verify their correctness essential. The
simulation of stabilizer circuits, which aims to replicate their behavior using
a classical computer, is known to be efficient and provides a means of testing
correctness. However, simulation is limited in its ability to examine the
exponentially large space of possible measurement outcomes. We propose a
comprehensive set of efficient classical algorithms to fully characterize and
exhaustively verify stabilizer circuits with Pauli unitaries conditioned on
parities of measurements. We introduce, as a practical characterization, a
general form for such circuits and provide an algorithm to find a general form
of any stabilizer circuit. We then provide an algorithm for checking the
equivalence of stabilizer circuits. When circuits are not equivalent our
algorithm suggests modifications for reconciliation. Next, we provide an
algorithm that characterizes the logical action of a (physical) stabilizer
circuit on an encoded input. All of our algorithms provide relations of
measurement outcomes among corresponding circuit representations. Finally, we
provide an analytic description of the logical action induced by measuring a
stabilizer group, with application in correctness proofs of code-deformation
protocols including lattice surgery and code switching.Comment: 90 pages,21 figure
Shorter quantum circuits
We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the works arXiv:1612.01011 and arXiv:1612.02689, we show that taking probabilistic mixtures of channels to solve fallback (arXiv:1409.3552) and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+ gate set we achieve an average non-Clifford gate count of 0.23log2(1/)+2.13 and T-count 0.56log2(1/)+5.3 with mixed fallback approximations for diamond norm accuracy .
This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+). We also provide detailed numerical results for Clifford+T and Clifford+ gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices
Shorter quantum circuits via single-qubit gate approximation
We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the works \cite{Hastings2017} and \cite{Campbell2017}, we show that taking probabilistic mixtures of channels to solve fallback \cite{BRS2015} and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+ gate set we achieve an average non-Clifford gate count of and T-count with mixed fallback approximations for diamond norm accuracy .
This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+). We also provide detailed numerical results for Clifford+T and Clifford+ gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices
Advances in compilation for quantum hardware -- A demonstration of magic state distillation and repeat-until-success protocols
Fault-tolerant protocols enable large and precise quantum algorithms. Many
such protocols rely on a feed-forward processing of data, enabled by a hybrid
of quantum and classical logic. Representing the control structure of such
programs can be a challenge. Here we explore two such fault-tolerant
subroutines and analyze the performance of the subroutines using Quantum
Intermediate Representation (QIR) as their underlying intermediate
representation. First, we look at QIR's ability to leverage the LLVM compiler
toolchain to unroll the quantum iteration logic required to perform magic state
distillation on the quantum error-correcting code as originally
introduced by Bravyi and Kitaev [Phys. Rev. A 71, 022316 (2005)]. This allows
us to not only realize the first implementation of a real-time magic state
distillation protocol on quantum hardware, but also demonstrate QIR's ability
to optimize complex program structures without degrading machine performance.
Next, we investigate a different fault-tolerant protocol that was first
introduced by Paetznick and Svore [arXiv:1311.1074 (2013)], that reduces the
amount of non-Clifford gates needed for a particular algorithm. We look at four
different implementations of this two-stage repeat-until-success algorithm to
analyze the performance changes as the results of programming choices. We find
the QIR offers a viable representation for a compiled high-level program that
performs nearly as well as a hand-optimized version written directly in quantum
assembly. Both of these results demonstrate QIR's ability to accurately and
efficiently expand the complexity of fault-tolerant protocols that can be
realized today on quantum hardware