743 research outputs found
Gapped and gapless phases of frustration-free spin-1/2 chains
We consider a family of translation-invariant quantum spin chains with
nearest-neighbor interactions and derive necessary and sufficient conditions
for these systems to be gapped in the thermodynamic limit. More precisely, let
be an arbitrary two-qubit state. We consider a chain of qubits with
open boundary conditions and Hamiltonian which is defined as the
sum of rank-1 projectors onto applied to consecutive pairs of qubits. We
show that the spectral gap of is upper bounded by if the
eigenvalues of a certain two-by-two matrix simply related to have equal
non-zero absolute value. Otherwise, the spectral gap is lower bounded by a
positive constant independent of (depending only on ). A key
ingredient in the proof is a new operator inequality for the ground space
projector which expresses a monotonicity under the partial trace. This
monotonicity property appears to be very general and might be interesting in
its own right. As an extension of our main result, we obtain a complete
classification of gapped and gapless phases of frustration-free
translation-invariant spin-1/2 chains with nearest-neighbor interactions.Comment: v3: published versio
Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets
We generalize an efficient exact synthesis algorithm for single-qubit
unitaries over the Clifford+T gate set which was presented by Kliuchnikov,
Maslov and Mosca. Their algorithm takes as input an exactly synthesizable
single-qubit unitary--one which can be expressed without error as a product of
Clifford and T gates--and outputs a sequence of gates which implements it. The
algorithm is optimal in the sense that the length of the sequence, measured by
the number of T gates, is smallest possible. In this paper, for each positive
even integer we consider the "Clifford-cyclotomic" gate set consisting of
the Clifford group plus a z-rotation by . We present an
efficient exact synthesis algorithm which outputs a decomposition using the
minimum number of z-rotations. For the Clifford+T case
the group of exactly synthesizable unitaries was shown to be equal to the group
of unitaries with entries over the ring .
We prove that this characterization holds for a handful of other small values
of but the fraction of positive even integers for which it fails to hold is
100%.Comment: v2: published versio
Universal adiabatic quantum computation via the space-time circuit-to-Hamiltonian construction
We show how to perform universal adiabatic quantum computation using a
Hamiltonian which describes a set of particles with local interactions on a
two-dimensional grid. A single parameter in the Hamiltonian is adiabatically
changed as a function of time to simulate the quantum circuit. We bound the
eigenvalue gap above the unique groundstate by mapping our model onto the
ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin
chain was computed exactly by Koma and Nachtergaele using its -deformed
version of SU(2) symmetry. We also discuss a related time-independent
Hamiltonian which was shown by Janzing to be capable of universal computation.
We observe that in the limit of large system size, the time evolution is
equivalent to the exactly solvable quantum walk on Young's lattice
Complexity of the XY antiferromagnet at fixed magnetization
We prove that approximating the ground energy of the antiferromagnetic XY
model on a simple graph at fixed magnetization (given as part of the instance
specification) is QMA-complete. To show this, we strengthen a previous result
by establishing QMA-completeness for approximating the ground energy of the
Bose-Hubbard model on simple graphs. Using a connection between the XY and
Bose-Hubbard models that we exploited in previous work, this establishes
QMA-completeness of the XY model
Correlation length versus gap in frustration-free systems
Hastings established exponential decay of correlations for ground states of gapped quantum many-body systems. A ground state of a (geometrically) local Hamiltonian with spectral gap ε has correlation length ξ upper bounded as ξ=O(1/ε). In general this bound cannot be improved. Here we study the scaling of the correlation length as a function of the spectral gap in frustration-free local Hamiltonians, and we prove a tight bound ξ=O(1/√ε) in this setting. This highlights a fundamental difference between frustration-free and frustrated systems near criticality. The result is obtained using an improved version of the combinatorial proof of correlation decay due to Aharonov, Arad, Vazirani, and Landau
A Compressed Classical Description of Quantum States
We show how to approximately represent a quantum state using the square root of the usual amount of classical memory. The classical representation of an n-qubit state psi consists of its inner products with O(sqrt{2^n}) stabilizer states. A quantum state initially specified by its 2^n entries in the computational basis can be compressed to this form in time O(2^n poly(n)), and, subsequently, the compressed description can be used to additively approximate the expectation value of an arbitrary observable. Our compression scheme directly gives a new protocol for the vector in subspace problem with randomized one-way communication complexity that matches (up to polylogarithmic factors) the optimal upper bound, due to Raz. We obtain an exponential improvement over Raz\u27s protocol in terms of computational efficiency
- …