# 3. Quantum computation using a single photon and linear optics

In the following sections, we explain past and recent progress in experiments on photonic quantum information. The first example is the experimental demonstration of quantum algorithms using a single photon and linear optics.

Quantum computation is a new concept that utilizes quantum superposition states for ultrafast parallel processing.^{18)} There have been several proposals for the realization of quantum computers. Of these, nuclear magnetic resonance quantum computation (NMR-QC) has played the role of a testbed for quantum algorithms.^{40)}^{,}^{41)} However, NMR-QC has the following restrictions: the input qubits are prepared in ‘hot mixed states’, and the results are always given by an average over a large number of quantum systems. Therefore, there is a class of algorithm that cannot be performed by NMR-QC.

Quantum computation using linear optics is an alternatively important testbed for quantum computing. If a single photon is used for computation, then the result is given not by an average, but by a single quantum computation. In quantum algorithms, appropriate unitary transformations are applied to quantum registers. Reck and Zeilinger^{42)} found that linear optics can be used to realize any unitary transformation. The theoretical proposal of quantum computation using linear optics was given by Takeuchi,^{43)} whereby the quantum register was realized by modes and polarization of photons, and the unitary transformation was implemented with linear optics. An example of an optical circuit for the 4-bit Deutsch-Jozsa quantum algorithm was presented, where for each quantum computation, the answer to the Deutsch-Jozsa problem is given by a single-photon detection. This is in striking contrast with the NMR quantum computer, where the results are always given by an average over a large number of quantum systems. A similar idea was suggested independently.^{44)}

We also reported an experimental demonstration of the Deutsch-Jozsa quantum computation algorithm using linear optics and a single photon.^{45)}^{,}^{46)} First, let us introduce the problem of the Deutsch-Jozsa algorithm. Suppose we are given 2*N* digits. We call the arrays “even” when they include as many 1’s as 0’s (*e.g.*, 1,0,1,0 for *N* = 2), and “uniform” when they are filled with only ‘0’s or ‘1’s (*e.g.*, 1,1,1,1). The problem for the Deutsch-Jozsa algorithm is to find the correct answer between “the given array is not even” and “the given array is not uniform”. When the array satisfies both cases, either of them can be the answer. A classical computer requires *N* + 1 steps in the worst case. However, a quantum computer can find the answer with *O*(log(*N*)) steps.^{19)}

In the experiment (Fig. 3), an incident single photon becomes a superposition of being in four optical paths (Path 1 to Path 4). The polarization of the state at each path is then rotated by electro-optic (E/O) modulators upon the oracle function *f*(*i*), of which the value is either 0 or 1. When the photon is detected by the photon detector, the answer is that the given oracle *f*(*i*) is not even. If the photon is not detected, then the answer is that the given oracle *f*(*i*) is not uniform. For the input single photons, strongly attenuated light from a laser diode (LD) was used, where the average number of photons present in the optical system was 3 × 10^{−3}, and the probability of finding two or more photons in the coherent length of 10 cm was less than 3 × 10^{−4}. In this sense, the computation was performed using the quantum phenomenon of single-photon interference. A reference light was used for precise control of the path lengths.

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| Fig. 3. Schematic diagram of the optical system used for the Deutsch-Jozsa algorithm with 4-bit inputs. Reprinted with permission from Shigeki Takeuchi, Physical Review A, Vol. 62, 032301, 2000, Copyright (2000) by the American Physical Society. |

The experimental results are shown in Fig. 4. The vertical axis shows the four-bit digits given to the quantum computer and the horizontal axis shows the probability *P* of photon detection at the output port. The theoretical values are shown by the solid lines and the experimental values are plotted as black dots. This result shows that we can determine whether the statement “the given oracle *f*(*i*) is not even” or “the given oracle *f*(*i*) is not uniform” is correct with small average error rates of 2.7% and 4.0%, respectively, by the detection of a single photon. Here, the initial state was a pure state and the answer was given by single photon detection. Thus, the key aspect of the Deutsch-Jozsa algorithm to obtain the answer with *a single quantum computation* was fully demonstrated for the first time. The experiment was equivalent to 3 qubits, which was the largest size of a quantum computer when reported. Our demonstration suggested that quantum computation using linear optics is not only useful as a testbed for quantum algorithms, but also practical for small scale quantum processing, namely for quantum communication.

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| Fig. 4. Photon detection probability for the given 4-bit digits f(i). The theoretical values without errors are shown by solid lines and the experimental values are plotted as black dots. Reprinted with permission from Shigeki Takeuchi, Physical Review A, Vol. 62, 032301, 2000, Copyright (2000) by the American Physical Society. |

Although this scheme is useful to implement small scale quantum processing, there is a problem in that the number of required modes increases exponentially as the number of qubits increase. To overcome this difficulty, multiple photons must be used for multiple qubits and realize two-qubit gates between single photon inputs, which is discussed in the next section.