Measurement-based quantum computing (MBQC) is a model of quantum computing that proceeds by sequential measurements of individual spins in an entangled resource state. However, it remains a challenge to produce efficiently such resource states. Would it be possible to generate these states by simply cooling a quantum many-body system to its ground state? Cluster states, the canonical resource states for MBQC, do not occur naturally as unique ground states of physical systems. This inherent hurdle has led to a significant effort to identify alternative resource states that appear as ground states in spin lattices. Recently, some interesting candidates have been identified with various valence-bond-solid (VBS) states. In this review, we provide a pedagogical introduction to recent progress regarding MBQC with VBS states as possible resource states. This study has led to an interesting interdisciplinary research area at the interface of quantum information science and condensed matter physics.
We study a system of
N interacting triplons (the elementary excitations of a valence-bond solid) described by an effective interacting boson model derived within the bond operator formalism in order to determine the stability and the properties of many-triplon states. In particular, we consider the square lattice spin-$1/2\phantom{\rule{4pt}{0ex}}{J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Heisenberg model, focus on the intermediate parameter region where a quantum paramagnetic phase sets in, and consider the columnar valence-bond solid as a reference state. Within the bond operator theory, the Heisenberg model is mapped into an effective boson model in terms of triplet operators
t. The effective boson model is studied at the harmonic approximation and the energy of the triplons and the expansion of the triplon operators
b in terms of the triplet operators
t are determined. Such an expansion allows us to perform a second mapping and, therefore, determine an effective interacting boson model in terms of the triplon operators
b. We then consider systems with a fixed number
N of triplons and study the stability of many-triplon states within a mean-field approximation. We show that many-triplon states are stable, the lowest-energy ones are constituted by a small number of triplons, and the excitation gaps are finite. For
J2=0.48J1 and
0.52J1, we also calculate spin-spin and dimer-dimer correlation functions, dimer order parameters, and the bipartite von Neumann entanglement entropy within our mean-field formalism in order to determine the properties of the many-triplon state as a function of the triplon number
N. We find that the spin and the dimer correlations decay exponentially and that the entanglement entropy obeys an area law, regardless of the triplon number
N. Moreover, only for
J2=0.48J1, the spin correlations indicate that the many-triplon states with large triplon number
N might display a more homogeneous singlet pattern than the columnar valence-bond solid. We also comment on possible relations between the many-triplon states with large triplon number
N and gapped spin-liquid states.
Quantum computation is a novel way of information processing that allows, for certain classes of problems, exponential speedups over classical computation. Various models of quantum computation exist, such as the adiabatic, circuit, and measurement-based models. They have been proven equivalent in their computational power, but operate very differently. As such, they may be suitable for realization in different physical systems, and also offer different perspectives on open questions such as the precise origin of the quantum speedup. Here, we give an introduction to the one-way quantum computer, a scheme of measurement-based quantum computation (MBQC). In this model, the computation is driven by local measurements on a carefully chosen, highly entangled state. We discuss various aspects of this computational scheme, such as the role of entanglement and quantum correlations. We also give examples for ground states of simple Hamiltonians that enable universal quantum computation by local measurements.
In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The choice of basis for later measurements may depend on earlier measurement outcomes and the final result of the computation is determined from the classical data of all the measurement outcomes. This is in contrast to the more familiar gate array model in which computational steps are unitary operations, developing a large entangled state prior to some final measurements for the output. Two principal schemes of measurement based computation are teleportation quantum computation (TQC) and the so-called cluster model or one-way quantum computer (1WQC). We will describe these schemes and show how they are able to perform universal quantum computation. We will outline various possible relationships between the models which serve to clarify their workings. We will also discuss possible novel computational benefits of the measurement based models compared to the gate array model, especially issues of parallelisability of algorithms.
We consider the
S=1/2 Heisenberg model with nearest-neighbor interaction
J and an additional multispin interaction
Q3 on the square lattice. The
Q3 term consists of three bond-singlet projectors and is chosen to favor the formation of a valence-bond solid (VBS) where the valence bonds (singlet pairs) form a staggered pattern. The model exhibits a quantum phase transition from the N\'eel state to the VBS as a function of
Q3/J. We study the model using quantum Monte Carlo (stochastic series expansion) simulations. The N\'eel-VBS transition in this case is strongly first order in nature, in contrast to similar previously studied models with continuous transitions into columnar VBS states. The qualitatively different transitions illustrate the important role of an emerging
U(1) symmetry in the latter case, which is not possible in the present model due to the staggered VBS pattern (which does not allow local fluctuations necessary to rotate the local VBS order parameter).