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arXiv:quant-ph/0508048 · Physical Review Letters

Measuring the Parity of an N -Qubit State

Bei Zeng, Dong Zhou, Lixing You

Published Sep 8, 2005• 15 citations
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Abstract

We present a scheme for a projective measurement of the parity operator Pz = Product(N)i=1 sigma(i)z of N qubits. Our protocol uses a single ancillary qubit, or a probe qubit, and involves manipulations of the total spin of the N qubits without requiring individual addressing. We illustrate our protocol in terms of an experimental implementation with atomic ions in a two-zone linear Paul trap, and further discuss its extensions to several more general cases.

DOI
10.1103/physrevlett.95.110502

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We present a scheme for a projective measurement of the parity operator Pz = Product(N)i=1 sigma(i)z of N qubits. Our protocol uses a single ancillary qubit, or a probe qubit, and involves manipulations of the total spin of the N qubits without requiring individual addressing. We illustrate our protocol in terms of an experimental implementation with atomic ions in a two-zone linear Paul trap, and further discuss its extensions to several more general cases.
It is known that parity measurement, together with single-qubit rotation, is sufficient for implementing scalable quantum computation. In this Brief Report, we propose a scheme for a projective measurement of the parity operator ${P}_{z}={\ensuremath{\bigotimes}}_{i=1}^{N}{\ensuremath{\sigma}}_{i,z}$ of NNN distant atoms trapped in spatially separated cavities. Instead of direct interaction between the atoms, quantum interference of polarized photons decaying from the optical cavities is used to realize expected measurement without resorting to a sequence of single- and two-qubit operations. It is shown that parity measurement can be implemented repeatedly until success without destroying the qubits at any stage of the operation.
Effective spin-spin interactions between N+1 qubits enable the determination of the eigenvalue of an arbitrary Pauli product of dimension N with a constant, small number of multi-qubit gates that is independent of N and encodes the eigenvalue in the measurement basis states of an extra ancilla qubit. Such interactions are available whenever qubits can be coupled to a shared harmonic oscillator, a situation that can be realized in many physical qubit implementations. For example, suitable interactions have already been realized for up to 14 qubits in ion traps. It should be possible to implement stabilizer codes for quantum error correction with a constant number of multi-qubit gates, in contrast to typical constructions with a number of two-qubit gates that increases as a function of N. The special case of finding the parity of N qubits only requires a small number of operations that is independent of N. This compares favorably to algorithms for computing the parity on conventional machines, which implies a genuine quantum advantage.
We propose a protocol and physical implementation for partial Bell-state measurements of Fermionic qubits, allowing for deterministic quantum computing in solid-state systems without the need for two-qubit gates. Our scheme consists of two spin qubits in a double quantum dot where the two dots have different Zeeman splittings and resonant tunneling between the dots is only allowed when the spins are antiparallel. This converts spin parity into charge information by means of a projective measurement and can be implemented with established technologies. This measurement-based qubit scheme greatly simplifies the experimental realization of scalable quantum computers in electronic nanostructures.
This thesis reports high-fidelity near-field spatial microwave addressing of long-lived 43Ca+ "atomic clock" qubits performed in a two-zone single-layer surface-electrode ion trap. Addressing is implemented by using two of the trap's integrated microwave electrodes, one in each zone, to drive single-qubit rotations in the zone we choose to address whilst interferometrically cancelling the microwave field at the neighbour (non-addressed) zone. Using this field-nulling scheme, we measure a Rabi frequency ratio between addressed and non-addressed zones of up to 1400, from which we calculate an addressing error (or a spin-flip probability on the qubit transition) of 1e-6. Off-resonant excitation out of the qubit state is a more significant source of error in this experiment, but we also demonstrate polarisation control of the microwave field at an error level of 2e-5, which, if combined with individual-ion addressing, would be sufficient to suppress off-resonant excitation errors to the 1e-9 level. Further, this thesis presents preliminary results obtained with a micron-scale coupled-microstrip differential antenna probe that can be scanned over an ion-trap chip to map microwave magnetic near fields. The probe is designed to enable the measurement of fields at tens of microns above electrode surfaces and to act as an effective characterisation tool, speeding up design-fabrication-characterisation cycles in the production of new prototype microwave ion-trap chips. Finally, a new multi-layer design for an ion-trap chip which displays, in simulations, a 100-fold improvement in addressing performance, is presented. The chip electrode structure is designed to use the cancelling effect of microwave return currents to produce Rabi frequency ratios of order 1000 between trap zones using a single microwave electrode (i.e. without the need for nulling fields). If realised, this chip could be used to drive individually addressed single-qubit operations on arrays of memory qubits in parallel and with high fidelity.