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arXiv:1711.00094 · Physical Review A

Universal quantum computing using (Zd)3 symmetry-protected topologically ordered states

Yanzhu Chen, Abhishodh Prakash, Tzu-Chieh Wei

Published Feb 5, 2018• 22 citations
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Abstract

Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically nontrivial, short-ranged entangled states are promising candidates for such a resource. Miller and Miyake [npj Quantum Inf. 2, 16036 (2016)] recently constructed a particular ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-protected topological state on the Union Jack lattice and established its quantum-computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting $(d\ensuremath{-}1)$ qudit nontrivial ${\mathbb{Z}}_{d}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{d}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{d}$ symmetry-protected topological states are universal on the triangular lattice, for $d$ being a prime number greater than 2. The same construction also holds for other 3-colorable lattices, including the Union Jack lattice.

DOI
10.1103/physreva.97.022305

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Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically nontrivial, short-ranged entangled states are promising candidates for such a resource. Miller and Miyake [npj Quantum Inf. 2, 16036 (2016)] recently constructed a particular Z2\ifmmode×\else\texttimes\fiZ2\ifmmode×\else\texttimes\fiZ2{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​\ifmmode×\else\texttimes\fiZ2​ symmetry-protected topological state on the Union Jack lattice and established its quantum-computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting (d\ensuremath−1)(d\ensuremath{-}1)(d\ensuremath−1) qudit nontrivial Zd\ifmmode×\else\texttimes\fiZd\ifmmode×\else\texttimes\fiZd{\mathbb{Z}}_{d}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{d}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{d}Zd​\ifmmode×\else\texttimes\fiZd​\ifmmode×\else\texttimes\fiZd​ symmetry-protected topological states are universal on the triangular lattice, for ddd being a prime number greater than 2. The same construction also holds for other 3-colorable lattices, including the Union Jack lattice.
Measurement-based quantum computation is a model for quantum information processing utilizing local measurements on suitably entangled resource states for the implementation of quantum gates. A complete characterization for universal resource states is still missing. It has been shown that symmetry-protected topological order in one dimension can be exploited for the protection of certain quantum gates in measurement-based quantum computation. In this paper we show that the two-dimensional (2D) plaquette states on arbitrary lattices exhibit nontrivial symmetry-protected topological order in terms of symmetry fractionalization and that they are universal resource states for quantum computation. Our results of the nontrivial symmetry-protected topological order on arbitrary 2D lattices are based on an extension of the recent construction by Chen et al. [Phys. Rev. B 87, 155114 (2013)] on the square lattice.
While quantum computers can achieve dramatic speedups over the classical computers familiar to us, identifying the origin of this quantum advantage in physical systems remains a major goal of quantum information science. A useful tool here is measurement-based quantum computation (MQC), a computational framework utilizing the quantum entanglement found in many-body resource states. Not all resource states are useful for quantum computation however, so an important question is what properties of many-body entanglement characterize universal resource states, which can implement any quantum computation. Many-body states are also studied in condensed matter physics, where the collective behavior of quantum many-body systems sometimes define topological phases of matter. These phases are defined by nonlocal many-body entanglement, making topologically-ordered states natural candidates for MQC. We might wonder if these topological phases could be organized as phases of quantum computation, so that every state within the phase is universal for MQC. While phases of symmetry-protected topological order (SPTO) have arisen as natural candidates, previous attempts to demonstrate an MQC-SPTO correspondence were mostly limited to nonuniversal 1D spin chains, leaving the important 2D setting wide open. In this dissertation, we explore the wide and varied connections between MQC and SPTO, and obtain new results for 1D and 2D systems. After identifying a new MQC-SPTO correspondence within 1D spin chains, we move up and explore the operational use of 2D states with two complementary forms of SPTO. We create a new Union Jack resource state, whose different form of SPTO than previous 2D resource states permits a hierarchical notion of MQC universality. This state leads us to consider an idealized model of 2D SPTO, where we show that an additional symmetry condition makes these model states form universal resources for MQC only when they have nontrivial SPTO. We finally study the intrinsic complexity of SPTO-inspired states for classically intractable sampling, and identify inherent advantages of MQC for this purpose. Our work highlights the rich complexity available in states of entangled quantum matter, providing new evidence which sharpens our understanding of the diverse connections between MQC and SPTO.
A cluster state is a highly entangled quantum state that serves as a universal resource for measurement-based quantum computation. It is generated by applying controlled-Z (CZ) gates to the product state $\left\vert ++\cdots +\right\rangle $, and its parent Hamiltonian is the ZXZ model. This model exhibits a topological phase protected by the Z2even×Z2odd\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}Z2even​×Z2odd​\ symmetry. By applying general quantum gates to the state $\left\vert ++\cdots +\right\rangle ,wesystematicallyobtainageneralshort−rangeentangledclusterstate.Ifweusethecontrolled−controlledZ(CCZ)gateinsteadoftheCZgate,weobtainhyper−clusterstatesfeaturingfive−bodyinteractionterms.ThesestatesenabletheimplementationoftheCZgatewithinmeasurement−basedquantumcomputation.Motivatedbythis,weproposeageneralizationtotheC, we systematically obtain a general short-range entangled cluster state. If we use the controlled-controlled Z (CCZ) gate instead of the CZ gate, we obtain hyper-cluster states featuring five-body interaction terms. These states enable the implementation of the CZ gate within measurement-based quantum computation. Motivated by this, we propose a generalization to the C,wesystematicallyobtainageneralshort−rangeentangledclusterstate.Ifweusethecontrolled−controlledZ(CCZ)gateinsteadoftheCZgate,weobtainhyper−clusterstatesfeaturingfive−bodyinteractionterms.ThesestatesenabletheimplementationoftheCZgatewithinmeasurement−basedquantumcomputation.Motivatedbythis,weproposeageneralizationtotheC^{N}$Z gate, leading to a hyper-cluster model, in which (2N+1)(2N+1)(2N+1)-body entangled states are generated. This model retains the Z2even×Z2odd\mathbb{Z}_{2}^{\text{even}}\times \mathbb{Z}_{2}^{\text{odd}}Z2even​×Z2odd​\ symmetry, which lies outside the Clifford group for N≥3N\geq 3N≥3. We demonstrate that an open chain in this model exhibits a 22N2^{2N}22N fold degenerate ground states protected by the symmetry, corresponding to the emergence of NNN effective free spins at each edge. These boundary degrees of freedom can serve as NNN qubits for measurement-based quantum computation. In particular, this framework makes it possible to implement the CN−1^{N-1}N−1Z gate in measurement-based quantum computation.
Resource states that contain nontrivial symmetry-protected topological order are identified for universal single-qudit measurement-based quantum computation. Our resource states fall into two classes: one as the qudit generalizations of the one-dimensional qubit cluster state, and the other as the higher-symmetry generalizations of the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) state, namely, with unitary, orthogonal, or symplectic symmetry. The symmetry in cluster states protects information propagation (identity gate), while the higher symmetry in AKLT-type states enables nontrivial gate computation. This work demonstrates a close connection between measurement-based quantum computation and symmetry-protected topological order.
Recent progress in the characterization of gapped quantum phases has also triggered the search for a universal resource for quantum computation in symmetric gapped phases. Prior works in one dimension suggest that it is a feature more common than previously thought, in that nontrivial one-dimensional symmetry-protected topological (SPT) phases provide quantum computational power characterized by the algebraic structure defining these phases. Progress in two and higher dimensions so far has been limited to special fixed points. Here we provide two families of two-dimensional Z2{Z}_{2}Z2​ symmetric wave functions such that there exists a finite region of the parameter in the SPT phases that supports universal quantum computation. The quantum computational power appears to lose its universality at the boundary between the SPT and the symmetry-breaking phases.
Topological measurement-based quantum computation (MBQC) enables one to carry out universal fault-tolerant quantum computation via single-qubit measurements with a family of large entangled states called cluster states as resources. Raussendorf's three-dimensional cluster states (RTCSs) based on the surface codes are mainly considered for topological MBQC. In such schemes, however, the logical Hadamard, phase (Z1/2)({Z}^{1/2})(Z1/2), and TTT (Z1/4)({Z}^{1/4})(Z1/4) gates which are essential for building up arbitrary logical gates are not implemented natively without using state distillation or lattice dislocations, to the best of our knowledge. In particular, state distillation generally consumes many ancillary logical qubits; thus it is a severe obstacle against practical quantum computing. To solve this problem, we suggest an MBQC scheme via a family of cluster states called color-code-based cluster states (CCCSs) based on the two-dimensional color codes instead of the surface codes. We define logical qubits, construct elementary logical gates, and describe error correction schemes. We show that all logical Clifford gates generated by the cnot, Hadamard, and phase gates can be implemented natively in a fault-tolerant manner, although the TTT gate still requires state distillation to be fault-tolerant. The error thresholds of MBQC via CCCSs for logical-ZZZ errors are calculated to be 2.7%--2.8%, which are comparable to the values for RTCSs, assuming a simple error model where physical qubits have nontrivial errors independently with the same probability. We analyze and compare the resource overheads of both the schemes. In particular, we show that the number of physical qubits required for implementing a phase gate with CCCSs is at least about 26 times smaller than with RTCSs using state distillation, for the same code distance.