83 research outputs found
Matchgates and classical simulation of quantum circuits
Let G(A,B) denote the 2-qubit gate which acts as the 1-qubit SU(2) gates A
and B in the even and odd parity subspaces respectively, of two qubits. Using a
Clifford algebra formalism we show that arbitrary uniform families of circuits
of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines,
can be classically efficiently simulated. This reproduces a result originally
proved by Valiant using his matchgate formalism, and subsequently related by
others to free fermionic physics. We further show that if the n.n. condition is
slightly relaxed, to allowing the same gates to act only on n.n. and next-n.n.
qubit lines, then the resulting circuits can efficiently perform universal
quantum computation. From this point of view, the gap between efficient
classical and quantum computational power is bridged by a very modest use of a
seemingly innocuous resource (qubit swapping). We also extend the simulation
result above in various ways. In particular, by exploiting properties of
Clifford operations in conjunction with the Jordan-Wigner representation of a
Clifford algebra, we show how one may generalise the simulation result above to
provide further classes of classically efficiently simulatable quantum
circuits, which we call Gaussian quantum circuits.Comment: 18 pages, 2 figure
A diagrammatic calculus of fermionic quantum circuits
We introduce the fermionic ZW calculus, a string-diagrammatic language for
fermionic quantum computing (FQC). After defining a fermionic circuit model, we
present the basic components of the calculus, together with their
interpretation, and show how the main physical gates of interest in FQC can be
represented in our language. We then list our axioms, and derive some
additional equations. We prove that the axioms provide a complete equational
axiomatisation of the monoidal category whose objects are systems of finitely
many local fermionic modes (LFMs), with maps that preserve or reverse the
parity of states, and the tensor product as monoidal product. We achieve this
through a procedure that rewrites any diagram in a normal form. As an example,
we show how the statistics of a fermionic Mach-Zehnder interferometer can be
calculated in the diagrammatic language. We conclude by giving a diagrammatic
treatment of the dual-rail encoding, a standard method in optical quantum
computing used to perform universal quantum computation
The Computational Power of Non-interacting Particles
Shortened abstract: In this thesis, I study two restricted models of quantum
computing related to free identical particles.
Free fermions correspond to a set of two-qubit gates known as matchgates.
Matchgates are classically simulable when acting on nearest neighbors on a
path, but universal for quantum computing when acting on distant qubits or when
SWAP gates are available. I generalize these results in two ways. First, I show
that SWAP is only one in a large family of gates that uplift matchgates to
quantum universality. In fact, I show that the set of all matchgates plus any
nonmatchgate parity-preserving two-qubit gate is universal, and interpret this
fact in terms of local invariants of two-qubit gates. Second, I investigate the
power of matchgates in arbitrary connectivity graphs, showing they are
universal on any connected graph other than a path or a cycle, and classically
simulable on a cycle. I also prove the same dichotomy for the XY interaction.
Free bosons give rise to a model known as BosonSampling. BosonSampling
consists of (i) preparing a Fock state of n photons, (ii) interfering these
photons in an m-mode linear interferometer, and (iii) measuring the output in
the Fock basis. Sampling approximately from the resulting distribution should
be classically hard, under reasonable complexity assumptions. Here I show that
exact BosonSampling remains hard even if the linear-optical circuit has
constant depth. I also report several experiments where three-photon
interference was observed in integrated interferometers of various sizes,
providing some of the first implementations of BosonSampling in this regime.
The experiments also focus on the bosonic bunching behavior and on validation
of BosonSampling devices. This thesis contains descriptions of the numerical
analyses done on the experimental data, omitted from the corresponding
publications.Comment: PhD Thesis, defended at Universidade Federal Fluminense on March
2014. Final version, 208 pages. New results in Chapter 5 correspond to
arXiv:1106.1863, arXiv:1207.2126, and arXiv:1308.1463. New results in Chapter
6 correspond to arXiv:1212.2783, arXiv:1305.3188, arXiv:1311.1622 and
arXiv:1412.678
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