82 research outputs found
Complexity classification of two-qubit commuting hamiltonians
We classify two-qubit commuting Hamiltonians in terms of their computational
complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can
apply to any pair of qubits, starting in a computational basis state. We prove
a dichotomy theorem: either this model is efficiently classically simulable or
it allows one to sample from probability distributions which cannot be sampled
from classically unless the polynomial hierarchy collapses. Furthermore, the
only simulable Hamiltonians are those which fail to generate entanglement. This
shows that generic two-qubit commuting Hamiltonians can be used to perform
computational tasks which are intractable for classical computers under
plausible assumptions. Our proof makes use of new postselection gadgets and Lie
theory.Comment: 34 page
Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup
At its core a -design is a method for sampling from a set of unitaries in
a way which mimics sampling randomly from the Haar measure on the unitary
group, with applications across quantum information processing and physics. We
construct new families of quantum circuits on -qubits giving rise to
-approximate unitary -designs efficiently in
depth. These quantum circuits are based on a relaxation of technical
requirements in previous constructions. In particular, the construction of
circuits which give efficient approximate -designs by Brandao, Harrow, and
Horodecki (F.G.S.L Brandao, A.W Harrow, and M. Horodecki, Commun. Math. Phys.
(2016).) required choosing gates from ensembles which contained inverses for
all elements, and that the entries of the unitaries are algebraic. We reduce
these requirements, to sets that contain elements without inverses in the set,
and non-algebraic entries, which we dub partially invertible universal sets. We
then adapt this circuit construction to the framework of measurement based
quantum computation(MBQC) and give new explicit examples of -qubit graph
states with fixed assignments of measurements (graph gadgets) giving rise to
unitary -designs based on partially invertible universal sets, in a natural
way. We further show that these graph gadgets demonstrate a quantum speedup, up
to standard complexity theoretic conjectures. We provide numerical and
analytical evidence that almost any assignment of fixed measurement angles on
an -qubit cluster state give efficient -designs and demonstrate a quantum
speedup.Comment: 25 pages,7 figures. Comments are welcome. Some typos corrected in
newest version. new References added.Proofs unchanged. Results unchange
Topological Quantum Compiling
A method for compiling quantum algorithms into specific braiding patterns for
non-Abelian quasiparticles described by the so-called Fibonacci anyon model is
developed. The method is based on the observation that a universal set of
quantum gates acting on qubits encoded using triplets of these quasiparticles
can be built entirely out of three-stranded braids (three-braids). These
three-braids can then be efficiently compiled and improved to any required
accuracy using the Solovay-Kitaev algorithm.Comment: 20 pages, 20 figures, published versio
Extending ancilla driven universal quantum computation beyond stepwise determinism
A major research goal in the field of quantum computation is the construction of the universal quantum computer (UQC): a device that can implement any quantum algorithm. Several theoretical schemes for implementing UQC have been developed which require different sets of resources and capabilities with varying implications for the optimum experimental implementations. The ancilla driven quantum computation scheme (ADQC) comprises two subsystems: a memory register of qubits on which information is retained and processed and an ancilla system of qubits which couple to the register. This coupling is represented in the ADQC scheme by a fixed quantum gate.By preparing the ancilla in selected states before applying this gate and then measuring it in selected measurement basis afterwards, quantum gates are enacted on the register qubits. ADQC is deterministic in that the probability of the outcome after performing the entire procedure is 1 but we have to apply corrections to the procedure at each step that depend on the probabilistic outcome of the ancilla measurement. An important resource in this model is the availability of a maximally entangling two-qubit gate between the ancilla and register qubits because if the gate is not maximally entangling,the resulting gates on the register can not be selected with stepwise determinism.It is proven in this thesis that in fact ADQC with non-maximally entangling interaction gates is universal. This requires showing that single- and two-qubit unitary gates can be effciently implemented probabilistically. We also show a relationship between the expected time of the probabilistic implementation of a gate and the ability to control the ancilla. In the ADQC model, the ancilla is controlled with single qubit unitary gates just before interacting with the register and just before measurement.We show that the increase in time caused by a loss of maximally entangling two-qubit gates can be counteracted by control over the ancilla. This needs not be the ability to perform any single qubit unitary to the ancilla but just the ability to perform a specific small finite set of operations.This is important because the resource requirements described by a scheme affect the properties of possible experimental implementations. The ADQC scheme was originally designed to be used with physical implementations of quantum computing that involves qubits coming from different physical systems that have different properties.This may restrict the availability of couplings between the register and ancilla systems equivalent to maximally entangling quantum gates. By further focusing on the model under specific restrictions, such as minimal control of the ancilla system or long distance separation between register qubits, we find certain properties of the physical implementation that may best suit it for ADQC beyond stepwise determinism. Minimal control appears best suited for symmetric ancilla-register interactions; use overlong distances suits a transmitter going to an unknown receiver with possible small errors in the receiver's interaction with the ancilla.A major research goal in the field of quantum computation is the construction of the universal quantum computer (UQC): a device that can implement any quantum algorithm. Several theoretical schemes for implementing UQC have been developed which require different sets of resources and capabilities with varying implications for the optimum experimental implementations. The ancilla driven quantum computation scheme (ADQC) comprises two subsystems: a memory register of qubits on which information is retained and processed and an ancilla system of qubits which couple to the register. This coupling is represented in the ADQC scheme by a fixed quantum gate.By preparing the ancilla in selected states before applying this gate and then measuring it in selected measurement basis afterwards, quantum gates are enacted on the register qubits. ADQC is deterministic in that the probability of the outcome after performing the entire procedure is 1 but we have to apply corrections to the procedure at each step that depend on the probabilistic outcome of the ancilla measurement. An important resource in this model is the availability of a maximally entangling two-qubit gate between the ancilla and register qubits because if the gate is not maximally entangling,the resulting gates on the register can not be selected with stepwise determinism.It is proven in this thesis that in fact ADQC with non-maximally entangling interaction gates is universal. This requires showing that single- and two-qubit unitary gates can be effciently implemented probabilistically. We also show a relationship between the expected time of the probabilistic implementation of a gate and the ability to control the ancilla. In the ADQC model, the ancilla is controlled with single qubit unitary gates just before interacting with the register and just before measurement.We show that the increase in time caused by a loss of maximally entangling two-qubit gates can be counteracted by control over the ancilla. This needs not be the ability to perform any single qubit unitary to the ancilla but just the ability to perform a specific small finite set of operations.This is important because the resource requirements described by a scheme affect the properties of possible experimental implementations. The ADQC scheme was originally designed to be used with physical implementations of quantum computing that involves qubits coming from different physical systems that have different properties.This may restrict the availability of couplings between the register and ancilla systems equivalent to maximally entangling quantum gates. By further focusing on the model under specific restrictions, such as minimal control of the ancilla system or long distance separation between register qubits, we find certain properties of the physical implementation that may best suit it for ADQC beyond stepwise determinism. Minimal control appears best suited for symmetric ancilla-register interactions; use overlong distances suits a transmitter going to an unknown receiver with possible small errors in the receiver's interaction with the ancilla
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