Measurement-Based Quantum Circuits Model

Osama M. Raisuddin, Suvranu De

Kai Klocke, Michael Buchhold

Projective measurements in random quantum circuits lead to a rich breadth of entanglement phases and extend the realm of nonunitary quantum dynamics.Here, we explore the connection between measurement-only quantum circuits in one spatial dimension and the statistical mechanics of loop models in two dimensions.While Gaussian Majorana circuits admit a microscopic mapping to loop models, for non-Gaussian, i.e., generic Clifford, circuits, a corresponding mapping may emerge only on a coarse-grained scale.We then focus on a fundamental symmetry of loop models: the orientability of worldlines.We discuss how orientability enters in the measurement framework, acting as a separatrix for the universal long-wavelength behavior in a circuit.When orientability is broken, the circuit falls into the universality class of closely packed loops with crossings (CPLC) and features a Goldstone phase with a peculiar, universal log 2 ðLÞ scaling of the entanglement entropy.In turn, when orientability is preserved, the longwavelength behavior of the circuit mimics that of (coupled) two-dimensional Potts models.We demonstrate the strength of the loop model approach by numerically simulating a variety of measurement-only Clifford circuits.Upon varying the set of measured operators, a rich circuit dynamics is observed, ranging from CPLC to the one-state Potts model (percolation), the two-state Potts model (Ising), and coupled Potts models (Berezhinskii-Kosterlitz-Thouless) universality class.Loop models, thus, provide a handle to access a large class of measurement-only circuits and yield a blueprint on how to realize desired entanglement phases by measurement.

N. E. Bonesteel, David Peter Divincenzo

We construct quantum circuits for measuring the commuting set of vertex and plaquette operators that appear in the Levin-Wen model for doubled Fibonacci anyons. Such measurements can be viewed as syndrome measurements for the quantum error-correcting code defined by the ground states of this model (the Fibonacci code). We quantify the complexity of these circuits with gate counts using different universal gate sets and find these measurements become significantly easier to perform if $n$-qubit Toffoli gates with $n=3,\phantom{\rule{0.28em}{0ex}}4$, and 5 can be carried out directly. In addition to measurement circuits, we construct simplified quantum circuits requiring only a few qubits that can be used to verify that certain self-consistency conditions, including the pentagon equation, are satisfied by the Fibonacci code.

Hans J. Briegel, Dan E. Browne, Wolfgang Dür +2 more

Phillip Kaye, Raymond Laflamme, Michele Mosca

In Section 1.3, we introduced the circuit model of (classical) computation. We restricted attention to reversible circuits since they can simulate any non-reversible circuit with modest overhead. This model can be generalized to a model of quantum circuits. In the quantum circuit model, we have logical qubits carried along ‘wires’, and quantum gates that act on the qubits. A quantum gate acting on n qubits has the input qubits carried to it by n wires, and n other wires carry the output qubits away from the gate. A quantum circuit is often illustrated schematically by a circuit diagram as shown in Figure 4.1. The wires are shown as horizontal lines, and we imagine the qubits propagating along the wires from left to right in time. The gates are shown as rectangular blocks. For convenience, we will restrict attention to unitary quantum gates (which are also reversible). Recall from Section 3.5.3 that non-unitary (non-reversible) quantum operations can be simulated by unitary (reversible) quantum gates if we allow the possibility of adding an ancilla and of discarding some output qubits. A circuit diagram describing a superoperator being implemented using a unitary operator is illustrated in Figure 4.2. In the example of Figure 4.1, the 4-qubit state |ψi⟩= |0⟩⊗ |0⟩⊗ |0⟩⊗ |0⟩ enters the circuit at the left (recall we often write this state as |ψi⟩ = |0⟩|0⟩|0⟩|0⟩ or |ψi⟩ = |0000⟩.) These qubits are processed by the gates U1, U2, U3, and U4. At the output of the circuit we have the collective (possibly entangled) 4-qubit state |ψf⟩. A measurement is then made of the resulting state. The measurement will often be a simple qubit-by-qubit measurement in the computational basis, but in some cases may be a more general measurement of the joint state. A measurement of a single qubit in the computational basis is denoted on a circuit diagram by a small triangle, as shown in Figure 4.1 (there are other symbols used in the literature, but we adopt this one). The triangle symbol will be modified for cases in which there is a need to indicate different types of measurements.R50

Margareta Vamos