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arXiv:1205.4740 · Physical Review Letters

Observation of Quantum Interference as a Function of Berry’s Phase in a Complex Hadamard Optical Network

Anthony Laing, Thomas Lawson, Enrique Martín López, Jeremy L. O’Brien

Published Jun 28, 2012• 13 citations
arXivHTMLPDF

Abstract

Emerging models of quantum computation driven by multiphoton quantum interference, while not universal, may offer an exponential advantage over classical computers for certain problems. Implementing these circuits via geometric phase gates could mitigate requirements for error correction to achieve fault tolerance while retaining their relative physical simplicity. We report an experiment in which a geometric phase is embedded in an optical network with no closed loops, enabling quantum interference between two photons as a function of the phase.

DOI
10.1103/physrevlett.108.260505

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Emerging models of quantum computation driven by multiphoton quantum interference, while not universal, may offer an exponential advantage over classical computers for certain problems. Implementing these circuits via geometric phase gates could mitigate requirements for error correction to achieve fault tolerance while retaining their relative physical simplicity. We report an experiment in which a geometric phase is embedded in an optical network with no closed loops, enabling quantum interference between two photons as a function of the phase.
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct π/8 and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases, or based on Aharonov-Anandan phases when the gyromagnetic ratios of the two qubits are equal. In implementing a universal set of quantum gates, one may combine geometric π/8 and Hadamard gates and dynamical gate.
To realize fault-tolerant quantum computing, it is necessary to store quantum information in logical qubits with error correction functions, realized by distributing a logical state among multiple physical qubits or by encoding it in the Hilbert space of a high-dimensional system. Quantum gate operations between these error-correctable logical qubits, which are essential for implementation of any practical quantum computational task, have not been experimentally demonstrated yet. Here we demonstrate a geometric method for realizing controlled-phase gates between two logical qubits encoded in photonic fields stored in cavities. The gates are realized by dispersively coupling an ancillary superconducting qubit to these cavities and driving it to make a cyclic evolution depending on the joint photonic state of the cavities, which produces a conditional geometric phase. We first realize phase gates for photonic qubits with the logical basis states encoded in two quasiorthogonal coherent states, which have important implications for continuous-variable-based quantum computation. Then we use this geometric method to implement a controlled-phase gate between two binomially encoded logical qubits, which have an error-correctable function.
We describe in detail a general strategy for implementing a conditional geometric phase between two spins. Combined with single-spin operations, this simple operation is a universal gate for quantum computation, in that any unitary transformation can be implemented with arbitrary precision using only single-spin operations and conditional phase shifts. Thus quantum geometrical phases can form the basis of any quantum computation. Moreover, as the induced conditional phase depends only on the geometry of the paths executed by the spins it is resilient to certain types of errors and offers the potential of a naturally fault-tolerant way of performing quantum computation.
In quantum information science, the phase of a wave function plays an important role in encoding information. Although most experiments in this field rely on dynamic effects to manipulate this information, an alternative approach is to use geometric phase, which has been argued to have potential fault tolerance. We demonstrated the controlled accumulation of a geometric phase, Berry's phase, in a superconducting qubit; we manipulated the qubit geometrically by means of microwave radiation and observed the accumulated phase in an interference experiment. We found excellent agreement with Berry's predictions and also observed a geometry-dependent contribution to dephasing.
Optical photons present many interesting aspects in the context of the research for suitable physical representations of quantum bits. Photons are chargeless particles, and do not interact strongly with each other, or even with most matter. They can be guided along long distances with low loss in optical fibers, delayed efficiently using phase shifters, and combined easily using beamsplitters. Photons exhibit typical quantum phenomena, such as the interference produced in two-slit experiments. Furthermore, in principle, photons can be made interact with each other if carefully handled. There are many practical problems with this approach; nevertheless it presents, beyond what has already been noted, an appealing conceptual simplicity.The purpose of this work is to present - at least, in its fundamental principles - a set of optical photon quantum gates that is universal for quantum computation. To do so, we begin by gathering the necessary mathematical and physical tools. We discuss canonical quantization for the electromagnetic field, and we formalize in an abstract context the characteristic properties of every realization of the Fock canonical commutation relationships (CCR). We analize the class of the spaces where it is possible to give such a realization, and that naturally lend themselves to host the physical model of our interest. Once acquired these bases, we turn ourselves to a particular optic system, the Mach-Zehnder interferometer, and we study the general properties of this system and of its essential constituents. At this point we choose a particular encoding of quantum bit and we look at the optical devices just discussed as elements of a quantum circuit. We then discuss the fundamental ideas that may guide the implementation of some important quantum gates: Hadamard gate, π/8\pi/8π/8 gate and controlled-NOT. We conclude by showing that these three quantum gates form a universal set of gates for quantum computation.
The Berry phase encodes the geometry of a closed Hamiltonian path, complementing the dynamical phase determined by the accumulated energy. We uncover an adiabatic error-cancellation principle that arises from this geometric character and has no counterpart for energy-based quantities. To extract the Berry phase while canceling the dynamical phase, we combine finite-runtime evolutions generated by ±H\pm H±H along the loop. This construction exactly cancels the leading O(T−1)O(T^{-1})O(T−1) and all higher odd-order nonoscillatory phase errors. Richardson extrapolation further reduces the leading residual error to an oscillatory contribution whose amplitude is set by the endpoints of the loop alone. Beyond this deterministic cancellation, runtime randomization suppresses the remaining oscillatory contribution, reducing the bias after rrr Richardson levels to O(T−(2r+2))O(T^{-(2r+2)})O(T−(2r+2)) for fixed rrr. The same average also suppresses the bias from an imperfectly prepared guiding state, so that the admissible preparation infidelity no longer depends on the target accuracy εB\varepsilon_BεB​. Combining these principles, we obtain a randomized Hadamard-test algorithm for Berry phase estimation that reduces the coherent evolution time to O(εB−1/(2r+2))O(\varepsilon_B^{-1/(2r+2)})O(εB−1/(2r+2)​) while retaining the standard O(εB−2)O(\varepsilon_B^{-2})O(εB−2​) sample complexity. This turns the geometric origin of the cancellation into an algorithmic advantage and makes Berry phase estimation a promising task for early fault-tolerant quantum computers.