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arXiv:0708.1062 · Physical Review A

Controlled-NOT gate for multiparticle qubits and topological quantum computation based on parity measurements

Oded Zilberberg, Bernd Braunecker, Daniel Loss

Published Jan 23, 2008• 53 citations
arXivHTMLPDF

Abstract

We discuss a measurement-based implementation of a controlled-NOT (CNOT) quantum gate. Such a gate has recently been discussed for free electron qubits. Here we extend this scheme for qubits encoded in product states of two (or more) spins 1/2 or in equivalent systems. The key to such an extension is to find a feasible qubit-parity meter. We present a general scheme for reducing this qubit-parity meter to a local spin-parity measurement performed on two spins, one from each qubit. Two possible realizations of a multiparticle CNOT gate are further discussed: electron spins in double quantum dots in the singlet-triplet encoding, and $\ensuremath{\nu}=5/2$ Ising non-Abelian anyons using topological quantum computation braiding operations and nontopological charge measurements.

DOI
10.1103/physreva.77.012327

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We discuss a measurement-based implementation of a controlled-NOT (CNOT) quantum gate. Such a gate has recently been discussed for free electron qubits. Here we extend this scheme for qubits encoded in product states of two (or more) spins 1/2 or in equivalent systems. The key to such an extension is to find a feasible qubit-parity meter. We present a general scheme for reducing this qubit-parity meter to a local spin-parity measurement performed on two spins, one from each qubit. Two possible realizations of a multiparticle CNOT gate are further discussed: electron spins in double quantum dots in the singlet-triplet encoding, and \ensuremathν=5/2\ensuremath{\nu}=5/2\ensuremathν=5/2 Ising non-Abelian anyons using topological quantum computation braiding operations and nontopological charge measurements.
Joint measurements of two-Pauli observables are powerful tools for both the control and protection of quantum information. By following a simple recipe for measurement choices, single- and two- qubit rotations using two-Pauli parity and single qubit measurements are guaranteed to be unitary while requiring only a single ancilla qubit. This language for measurement based quantum computing is shown to be directly applicable to encoded double quantum dot singlet-triplet spin qubits, by measuring spin-parity between dots from neighboring qubits. Along with exchange interaction, a complete, leakage free, measurement based gate set can be shown, up to a known Pauli correction. Both theoretically exact spin-parity measurements and experimentally demonstrated asymmetric spin-parity measurements are shown to be viable for achieving the proposed measurement based scheme, provided some extra leakage mitigating measurement steps. This method of spin qubit control offers a leakage suppressed, low resource overhead implementation of a measurement-based control that is viable on current spin qubit processor devices.
We extend the topological quantum computation scheme using the Pfaffian quantum Hall state, which has been recently proposed by Das Sarma et al., in a way that might potentially allow for the topologically protected construction of a universal set of quantum gates. We construct, for the first time, a topologically protected controlled-NOT gate, which is entirely based on quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the $\ensuremath{\pi}∕8$ gate, are also explicitly implemented by quasihole braidings. Instead of the $\ensuremath{\pi}∕8$ gate we try to construct a topologically protected Toffoli gate, in terms of the controlled-phase gate and CNOT or by a braid-group-based controlled-controlled-ZZZ precursor. We also give a topologically protected realization of the Bravyi-Kitaev two-qubit gate g3{g}_{3}g3​.
The quantum controlled-NOT (CNOT) gate is a prototypical two-qubit quantum logic gate that provides the basic controlled logic for a set of gates for universal quantum computation. It has been shown that parity checking devices can be used to construct CNOT gates, and the fidelity of a CNOT operation is highly constrained by the fidelity of parity detection with this strategy. In this paper, a scheme to implement a CNOT operation on two stationary electron spins confined in quantum dots (QDs) inside double-sided optical microcavities is presented, based on the faithful parity detection achieved by a heralded and robust two-electron-spin quantum nondemolition (QND) parity detector. The QND parity detector is considerably different from previous implementations and experimentally more realizable, and works in the heralded and repeat-until-success fashion with robust fidelity, which enables our CNOT gate to be implemented deterministically with unity fidelity. Moreover, based on the features of the QND parity detector, a complete Bell-state analysis on two QD-confined electron spins can be realized without wrong judgment or any destruction of the analyzed entangled state. The efficiency of parity detection is also discussed by considering currently achievable system parameters.
We extend the topological quantum computation scheme using the Pfaffian quantum Hall state, which has been recently proposed by Das Sarma et al., in a way that might potentially allow for the topologically protected construction of a universal set of quantum gates. We construct, for the first time, a topologically protected Controlled-NOT gate which is entirely based on quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the pi/8 gate, are also explicitly implemented by quasihole braidings. Instead of the pi/8 gate we try to construct a topologically protected Toffoli gate, in terms of the Controlled-phase gate and CNOT or by a braid-group based Controlled-Controlled-Z precursor. We also give a topologically protected realization of the Bravyi-Kitaev two-qubit gate g_3.
A crucial requirement for scalable quantum-information processing is the realization of multiple-qubit quantum gates. Universal multiple-qubit gates can be implemented by a set of universal single qubit gates and any one kind of two-qubit gate, such as a controlled-NOT (CNOT) gate. Semiconductor quantum dot qubits are a leading approach for the physical implementation of quantum computation, due to their potential for large-scale integration. Two-qubit gate operations have been so far only demonstrated in individual electron spin-based quantum dot systems. Due to the relatively short de-coherence time, charge qubits in quantum dots are generally considered to be inferior for going beyond single qubit level. Here, we demonstrate the benchmarking CNOT gate in two capacitively coupled charge qubits, each consisting of an electron confined in a GaAs/AlGaAs double quantum dot. Owing to the strong inter-qubit coupling strength, gate operations with a clock speed up to 5GHz has been realized. A processing tomography shows encouragingly that the universal two-qubit gate operations have comparable fidelities to that of spin-based two-qubit gates. Our results suggest that semiconductor charge qubits have a considerable potential for scalable quantum computing and may stimulate the use of long-range Coulomb interaction for coherent quantum control in other devices.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.