Quantum computing using single photons and the Zeno effect

We show that the quantum Zeno effect can be used to suppress the failure events that would otherwise occur in a linear optics approach to quantum computing. From a practical viewpoint, that would allow the implementation of deterministic logic gates without the need for ancilla photons or high-efficiency detectors. We also show that the photons can behave as if they were fermions instead of bosons in the presence of a strong Zeno effect, which leads to an alternative paradigm for quantum computation.

Figure 1

Implementation of a controlled-NOT logic gate that succeeds with probability $\frac{1}{4}$. All of the failure events correspond to situations in which two photons were emitted into the same optical fiber. A more detailed description can be found in Ref. 22.

Figure 2

Coupled optical fibers used to implement the beam splitting process in a continuous way, which allows the use of the Zeno effect. (a) Side view showing the coupling between two optical fiber cores via their evanescent field, which is equivalent to the tunneling of photons from one core to the other. (b) End view showing a dual-core optical fiber that could be used to implement such a beam splitter. Devices of this kind are commercially available.

Figure 3

(a) Operation of the $\sqrt{\mathrm{SWAP}}$ gate, which is defined in such a way that it swaps the values of any two logical inputs when it is applied twice (squared). If the operation is only applied once, however, a quantum superposition of states is created and neither qubit has a well-defined value, as illustrated by the question mark. (b) Potential implementation of a $\sqrt{\mathrm{SWAP}}$ gate by bringing the cores of two optical fibers in close proximity, which allows a photon in one core to be coupled into the other core. If length $L$ produces a SWAP operation, then half that length (dashed line) will produce the $\sqrt{\mathrm{SWAP}}$ operation, aside from errors that can be suppressed using the Zeno effect and a phase factor discussed in the text.

Figure 4

Probability that a single photon will be found in path 1 of the coupled optical fiber device of Figs. 2, 3. A single photon was assumed to be incident in path 1 and Schrödinger’s equation was solved using the Hamiltonian of Eq. (2). These results are equivalent to Rabi oscillations in a two-level atom.

Figure 5

Probability $P_E$ of an error in the output of the $\sqrt{{\mathrm{SWAP}}^{\prime }}$ gate when a photon is incident in both input modes. The dots correspond to the value of $P_E$ as a function of the number $N$ of equally-spaced measurements made to determine the presence of two photons in the same optical fiber core. The solid line corresponds to similar results obtained in the presence of two-photon absorption, where the parameter $N$ is then defined as $N=\Delta t∕4{\tau }_D$, with $\Delta t$ the interaction time and ${\tau }_D$ the two-photon decay time.

Figure 6

(a) Implementation of a SWAP operation for photons by simply crossing two optical fibers. (b) A controlled-$Z$ gate for photons constructed by applying the ${\mathrm{SWAP}}^{\prime }$ operation of Eq. (11) followed by the SWAP operation illustrated above. This circuit produces only the identity operator for noninteracting fermions.

Figure 7

Expanded view of one of the optical fiber cores containing a single atom. The cross section for the absorption of a resonant photon can be comparable to the area of the core itself, in which case there can be a large interaction between two photons and a single atom. The effects of detuning and collisions must also be taken into account, but those effects can be compensated to give a two-photon absorption rate comparable to that from this simple picture.

Figure 8

Energy-level diagrams illustrating the use of a three-level atom to produce two-photon absorption. The atomic levels are labeled 1 through 3 while the photons are represented by the arrows. (a) Successive absorption of two photons whose energies are on resonance with that of the atomic transitions. (b) Elimination of single-photon absorption by increasing the energy of level 2, in which case only two-photon absorption satisfies energy conservation. (c) A virtual process in which the two original photons with wave vector $k$ are absorbed, followed by the reemission of two different photons with wave vectors $k_1^{\prime }$ and $k_2^{\prime }$.

Figure 9

An alternative paradigm for quantum computation. It is assumed that a set of particles can be forced to behave like fermions while their paths are interchanged, which implements the ${\mathrm{SWAP}}^{\prime }$ operation of Eq. (11). The particles are then forced to behave like bosons while the paths are interchanged again, which implements the SWAP operation. The combination of these two operations implements a controlled-$Z$ gate (controlled phase gate), which is universal for quantum computation.

Figure 10

A two-qubit encoding that can be used to correct for failure events in which a Zeno effect of limited strength allows the absorption of two photons. Logical qubit $q$ is encoded in the values of two physical qubits (photons) $q_1$ and $q_2$, and similarly for logical qubit $q^{\prime }$.