Program Manager
Prof Andrew White - University of Queensland

Optical Quantum Computing Researchers
Staff
Dr Jeremy O'Brien
Dr Geoffrey Pryde
Dr Alexei Gilchrist
Dr Kenny Pregnell
Dr Kevin Resch

Students
Mr Rohan Dalton (PhD)
Mr Nathan Langford (PhD)
Mr Till Weinhold (PhD)

Collaborating Centre Researchers
Prof Gerard Milburn - University of Queensland
A/Prof Timothy Ralph - University of Queensland
Dr Daniel James - Los Alamos National Laboratory, USA
Dr Daniel James - University of Toronto, Canada

Other Collaborators
Dr William Munro - Hewlett Packard, UK
Prof Paul Kwiat - University of Illinois, USA
Dr Michael Harvey - University of Queensland
Mr Robert Prevedel - University of Vienna, Austria

Program Description
The aim of this program is to construct the basic building block of an optical quantum processor, focussing on the key 2-qubit CNOT gate, and to develop the foundations for a scaleable architecture. Our strategy involves both experimental and theoretical research, including: developing measurement techniques for characterising the relevant quantum states and processes; improving photon source and optical circuitry performance; development and application of measures of gate performance; and the realisation of simple quantum circuits. We are supported by ARC, US Government (DTO) and Queensland State Government funding.

1. Background
Modern optical technology allows very precise manipulation and measurement of
light; the basic particles of light, photons, experience very little intrinsic decoherence, as the electromagnetic environment at optical frequencies is a vacuum. These two features combine to make photonic qubits very appealing, as single qubits can be produced, manipulated and measured with low error rates. Quantum computation also requires that two photonic qubits be able to interact and influence one another, the typical example being via a controlled-NOT, or CNOT, gate. This cannot be achieved via normal nonlinear optical methods, as available materials produce interactions that are 10 million times weaker than required. However, in 2001 Knill, Laflamme & Milburn (KLM) proposed a scheme for efficient quantum computation with linear optics [1], where the necessary two-photon interaction can be achieved non-deterministically via measurement – the resulting gate can be made deterministic by embedding it in a teleporter.

In this scheme a single photon is used to encode each qubit. Attenuating a standard laser beam produces low average photon numbers, but this is not sufficient for quantum computation as we require exactly one photon in a given time period to avoid error. The current gold standard for photon sources is spontaneous parametric downconversion (SPDC), which produces photon pairs in well defined spatial and frequency modes, albeit non-deterministically. When used as a two-photon source, SPDC yields high rates, typically tens of thousands of photon pairs per second; when used as a four-photon source, the rates are much lower, typically a thousand counts per hour. High count rates are desirable both because they allow good statistical analysis in a reasonable time frame, and because they minimise the effect of long-term drift in the apparatus.

The original KLM proposal required an auxilliary, or ancillary, photon for each logical photonic qubit: thus a two-qubit gate required four photons. In 2001 groups from the University of Queensland [2] and Hokkaido University [3] published proposals to realise a CNOT gate with only two photons. These proposals suggested that comprehensive experimentation in linear optical quantum computing could be achieved using high count-rate (bright) two-photon sources.

The final stage in any quantum computation scheme is measuring the logical state, 0 or 1, of the output qubits. However during development of a quantum computer it is necessary to be able to fully characterise output states – and even better gate processes – to determine gate behaviour in terms of noise and entangling capability. Methods for measuring qubit states are now well developed: qubit state tomography with photonic qubits was demonstrated in [4], and a comprehensive theoretical analysis that allows for the effects of measurement uncertainty was given in [5].

In 2003 we constructed and observed quantum operation of a non-deterministic CNOT gate. Key design features were the use of polarisation displacers to produce a stable interferometric arrangement [6] and the use of wave-plates to produce beam mixing in a precise ratio. The operation of the gate was unambiguously quantum. This was determined by measuring the output density matrices for the logical-input data
(i.e. the |00>, |01>, |10>, |11> inputs), and, more significantly, for superposition
inputs – in the latter case the outputs are entangled. We measured both the fidelity of the output states with the ideal expected Bell-state outputs (e.g. 87% for the singlet
state |01> -|10>) and the tangle and linear entropy of the output states [7]. In 2004
we investigated important principles of characterising real-world quantum circuits, fully characterised our two-photon CNOT gate using quantum process tomography [8]. The CNOT gate, as well as being a key processing device in quantum computation, is also a key measurement device. At the simplest level it allows an ideal projective or quantum non-demolition (QND) measurement to be made on a single qubit.
We simply modified our gate so that the strength of the measurement was smoothly
varied from weak to strong [9,10]. This generalised measurement demonstrated that the QND measurement is coherent – a key requirement for quantum computation
applications.

2. Simplified entangling circuits
A key resource for using entanglement in quantum information protocols are
gates that are capable of entangling or disentangling qubits. Entangling gates lie at the heart of quantum computation protocols, and disentangling gates used in Bell state analysers are required for quantum teleportation. Conceptually, the simplest such two-qubit gate is the controlled-z (cz) gate, which in the logical basis produces a π phase shift on the |11> term, (i.e., |00>, |01>, |10>, |11> −> |00>, |01>, |10>, -|11>).
This is a maximally entangling gate, which when coupled with single qubit rotations is universal for quantum computing. (The more familiar CNOT gate is formed by applying a Hadamard gate H to the target qubit before and after a CZ gate.)

We experimentally demonstrated a nondeterministic linear-optics cz gate, and used
it as a Bell-state analyser. Our cz gate is the simplest entangling (or dis-entangling)
linear optics gate realized to date, requiring only three partially-polarising beam splitters (PPBSs), two half-wave plates, no classical interferometers, and no ancilla photons. It is non-deterministic and success is heralded by detection of a single photon in each of the outputs. We demonstrated the operation of this type of gate using photons generated both by continuous-wave (CW) and by femtosecond pulsed parametric downconversion – by comparing the two we find that temporal mode mismatch was not a significant factor in our gate’s performance. We fully characterised the operation in both regimes using quantum process tomography, and also demonstrated the use of this kind of gate for fully-resolving Bell measurements. Our simple entangling optical gate is promising for micro-optics or guided optics implementations where extremely good non-classical interference is realisable.

Figure 1 (a) Interferometric cz gate based on the approach in [7]. Gate operation is enabled by transforming each qubit from polarisation to spatial encoding, and back again. This requires high interferometric stability and spatio-temporal mode-matching for correct operation. (b) Partially-polarising beam splitter (PPBS) gate. Thequbits can remain polarisation-encoded, since thevertically-polarised modes are completely reflected by the first PPBS, and do not interact. Nonclassical interference occurs between the horizontally-polarised modes, with η=1/3. The subsequent PPBSs give the required losses in the cV and tV modes as shown in (a).

The best performing entangling gate implementations to date have been interferometric: a conceptual schematic of an interferometric optical cz gate, composed of three partially-reflecting beam splitters with reflectivity η=1/3, is shown in Figure 1(a). Each polarisation qubit input to the gate is split into two longitudinal spatial modes via a polarising beam splitter. Non-classical interference occurs between the horizontally-polarised modes at the 1/3 beamsplitter. With probability 1/9 the circuit performs the cz operation. After the network of 1/3 beamsplitters, the two spatial modes of the control and target must be recombined to return to polarisation encoded qubits.Since the phase relationship between the logical modes must be maintained throughout this operation, interferometric stability is required between the control and target modes. Inherently stable interferometers have previously been used [7,8] to achieve this – however these may not be suitable for scaling to large numbers in micro- or integrated- optical realisations. Our alternative approach does not require interferometric stability.

The experimental setup for the cz gate we have developed is shown schematically in
Figure 1(b). It employs partially polarising beamsplitters (PPBSs) that completely reflect vertically polarized light, and have a reflectance of 1/3 for horizontally polarized light.
As in Figure 1(a), only the H modes interfere nonclassically at the first PPBS, while the V modes each experience a 1/3 loss at the other two PPBSs: the half-wave plates between the first and subsequentPPBSs rotate polarisation by 90˚– a bit flip, or single qubit X gate – and effectively reverse the operation of the second two PPBSs. The circuit of Figure 1(b) therefore performs a cz gate with additional X gates on the control and target qubits. These additional X gates could be corrected by adding appropriate half-wave plates in the outputs, or by relabelling the logical states of the outputs – here we chose to relabel. The key advantage of the PPBS gate is that the polarisation modes are never spatially separated and recombined, and consequently no classical interference conditions are required. A single nonclassical interference at the first PPBS
is therefore the gate’s sole mode-matching condition.

To test multi-qubit circuits, multi-photon sources are required. The best known way to generate two or more photons is pulsed parametric downconversion: pump power densities far greater than those possible with CW sources lead to significantly higher probabilities of multi-photon events. However, the short pump pulses lead to a larger bandwidth of downconverted photons, and to more stringent temporal mode-matching requirements. Thus any new gate architecture should be shown to be compatible with both CW and pulsed sources. We tested the PPBS architecture with both CW and femtosecond-pulsed sources, which produce pairs of energy degenerate single photons via spontaneous parametric downconversion in a β-Barium- Borate (BBO) crystal. The gates were completely characterised via quantum process tomography [8].

A convenient representation of the measured process is the χ matrix, which is a complete and unique description of the process relative to a given basis of operators. The χ matrix for ideal CZ gate operation in the Pauli basis is shown in Figure 2(a) (all the components are real). The experimental results for the CW gate are shown in Figure 2(b), those for the pulsed gate in Figure 2(c). In the former case, it is clear that the major deviation from ideal operation is the larger than expected II term (0.36 instead of 0.25). This arises from imperfect mode matching at the first PPBS, resulting in imperfect nonclassical interference – effectively, the control and target qubits do not interfere and are simply transmitted through the circuit unchanged.

Gate performances can be quantified by calculating either the process fidelity, or the average gate fidelity, which is the fidelity between expected and actual output states, averaged over all pure inputs. The CW and pulsed gates have process fidelities of 74.6±0.3% and 84.0±0.1% respectively; and average gate fidelities of 79.7±0.2% and 87.2±0.1%. Even with the more stringent temporal mode-matching requirements, there was better gate operation in the pulsed regime. Our experimental set-up systematically induced fixed rotations of the input and output polarisations. For practical application of this gate, these rotations have no detrimental effect because they can be measured and compensated for by appropriate single qubit rotations. We gauged their effect by modelling arbitrary single qubit unitary operations on both control and target input and outputs: a numerical search identified polarisation corrections which increased the CW and pulsed process fidelities to 77.0±0.3% and 86.6±0.2% respectively; and average gate fidelities to 81.6±0.2% and 89.3±0.1%.

In summary, we proposed and demonstrated a new architecture for entangling optical gates. The key advantage of this new gate architecture is its simplicity and suitability for scaling – it requires only one nonclassical mode matching condition, and no classical interferometers. This is very promising for micro-optic and integrated-optic realisations of this gate, where extremely good mode matching can be expected.

	Figure 2 Quantum process tomography of the cz gate. Real components of the χ matrix for the: (a) ideal; (b) CW; and (c) pulsed CZ gate. The imaginary components of the experimental matrices are not shown: a few elements are on the order of 0.05, the majority on the order of 0.001.

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