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arXiv:1704.06504 · Quantum

Universal MBQC with generalised parity-phase interactions and Pauli measurements

Aleks Kissinger, John van de Wetering

Published Apr 26, 2019• 35 citations
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Abstract

We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form exp⁡(−iπ2nZ⊗Z) . When n=2 , these are equivalent, up to local Clifford unitaries, to graph states. However, when n>2 , their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli Z and X measurements and feed-forward. Even though, for n>2 , the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every n>2 this family is capable of producing all Clifford gates and all diagonal gates in the n -th level of the Clifford hierarchy.

DOI
10.22331/q-2019-04-26-134

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We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form exp⁡(−iπ2nZ⊗Z) . When n=2 , these are equivalent, up to local Clifford unitaries, to graph states. However, when n>2 , their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli Z and X measurements and feed-forward. Even though, for n>2 , the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every n>2 this family is capable of producing all Clifford gates and all diagonal gates in the n -th level of the Clifford hierarchy.
We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form exp⁡(−iπ2nZ⊗Z)\exp(-i\frac{\pi}{2^{n}} Z\otimes Z)exp(−i2nπ​Z⊗Z). When n=2n = 2n=2, these are equivalent, up to local Clifford unitaries, to graph states. However, when n>2n > 2n>2, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli ZZZ and XXX measurements and feed-forward. Even though, for n>2n > 2n>2, the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every n>2n > 2n>2 this family is capable of producing all Clifford gates and all diagonal gates in the nnn-th level of the Clifford hierarchy.
Measurement-based quantum computation (MBQC) is a strong contender for realizing quantum computers. A critical question for MBQC is the identification of resource graph states that can enable universal quantum computation. Any such universal family must have unbounded entanglement width, which is equivalent to the ability to produce any circle graph state from the states in the family using only local Clifford operations, local Pauli measurements, and classical communication. Yet, it was not previously known whether or not circle graph states themselves are a universal resource. We show that, in spite of their expressivity, circle graph states are not efficiently universal for MBQC (i.e., assuming BQP≠BPP\mathsf{BQP} \neq \mathsf{BPP}BQP=BPP). We prove this by articulating a precise graph-theoretic correspondence between circle graph states and a certain subset of fermionic Gaussian states. This is accomplished by synthesizing a variety of techniques that allow us to handle both stabilizer states and fermionic Gaussian states at the same time. As such, we anticipate that our developments may have broader applications beyond the domain of MBQC as well.
Circle graph states are a structurally important family of graph states. The family's entanglement is a priori high enough to allow for universal measurement-based quantum computation (MBQC); however, MBQC on circle graph states is actually efficiently classically simulable. In this work, we paint a detailed picture of the local equivalence of circle graph states. First, we consider the class of all graph states that are local unitary (LU)-equivalent to circle graph states. In graph-theoretical terms, this LU-equivalence class is the set of all graphs reachable from the family of circle graphs by applying rrr-local complementations. We prove that the only graph states that are LU-equivalent to circle graph states are circle graph states themselves: circle graphs are closed under rrr-local complementation. Second, we show that bipartite circle graph states, i.e., 2-colorable circle graph states, are in one-to-one correspondence with planar code states, on which MBQC is known to be efficiently classically simulable. Leveraging this correspondence, we present alternative, simple proofs that (1) if a planar code state is LU-equivalent to a stabilizer state, they are in fact local Clifford (LC)-equivalent to it and that (2) MBQC on all circle graph states is efficiently classically simulable. Third and finally, we demonstrate that the problem of counting the number of graph states LU-equivalent to a given graph state is #P\#\mathsf{P}#P-hard.
The direct compilation of algorithm-specific graph states in measurement-based quantum computation (MBQC) can lead to resource reductions in terms of circuit depth, entangling gates, and even the number of physical qubits. In this work, we extend previous studies on algorithm-tailored graph states to periodic sequences of generalized Pauli rotations, which commonly appear in, e.g., Trotterized Hamiltonian simulation. We first implement an enhanced simulated-annealing-based algorithm to find optimal periodic graph states within local-Clifford equivalent MBQC resources. In addition, we derive a novel scheme for the preparation of resource states based on a graph state and a ladder of CNOT gates, which we term anticommutation-based MBQC, since it uncovers a direct relationship between the graph state and the anticommutation matrix for the set of Hamiltonians generating the computation. We also deploy our two approaches to derive universal resource states from minimal universal sets of generating Hamiltonians. Finally, we demonstrate and compare both of our methods based on various examples from condensed matter physics and universal quantum computation.
Flow criteria are used to efficiently extract computations, either in the form of measurement patterns or quantum circuits, from ZX-diagrams. Existing criteria such as causal flow, generalised flow, and Pauli flow, were all originally formulated for graph states, so they require ZX-diagrams to be in a very particular graph-state-like form. This form is easily broken by applying basic ZX rules and makes establishing some desirable properties very complicated. Here, we introduce a new "ZX-native" flow criterion called ZX-flow, formulated using a new type of decoration of a ZX-diagram we call Pauli semiwebs. These are a generalisation of Pauli webs, which have recently been used extensively in reasoning about fault-tolerant computations in the ZX-calculus. We show that ZX-flow is straightforwardly preserved by all Clifford rewrites and furthermore that a ZX-diagram has ZX-flow if and only if it is Clifford-equivalent to a graph-like ZX-diagram with Pauli flow. Finally, we show that any diagram with ZX-flow can be readily interpreted either as a deterministic measurement-based computation or as a Clifford isometry followed by a sequence of Pauli exponentials. The latter can then be efficiently extracted to a quantum circuit.
We introduce an open-source software library Graphix, which optimizes and simulates measurement-based quantum computation (MBQC). By combining the measurement calculus with an efficient graph state simulator, Graphix allows the classical preprocessing of Pauli measurements in the measurement patterns, significantly reducing the number of operations required to perform the quantum computation while maintaining determinism. For a measurement pattern translated from a quantum circuit, this corresponds to the preprocessing of all Clifford gates, and this improvement in the one-way model is important for efficient operations in quantum hardware with limited qubit numbers. In addition to the direct translation from gate networks, we provide a pattern generation method based on flow-finding algorithms, which automatically generates byproduct correction sequences to ensure determinism. We further implement optimization strategies for measurement patterns beyond the standardization procedure and provide tensor-network backend for classically simulating the MBQC.