High precision voltage measurement for optical quantum computation

: We communicate the theoretical results in the study of utilizing the quantum phenomena 1 in optical current for quantum computation in the context of high precision voltage measurements. 2 These results can be used around computation by quantum sampling and quantum communication 3 as the basis for further research and physical implementations. We propose a Main Optical 4 Setup (MOS) for such computation which allows to make a superstructure to implement specific 5 computations and algorithms. To create MOS we used the nonlinear units (e.g., beta- barium 6 borate crystal), arranged in series, powered with pulsed laser pump, and ended with the beam 7 splitter to generate the output state of a number of entangled photon pairs. The computation is 8 made by propagation of entanglement with beam splitters applied crossword and adjustable phase 9 shifters that are tools for parameter steering. We show how to implement the series of cosine-based 10 components on the example of two-component case. The results opens a broad area for future 11 research in the area of building quantum optimizer using the quantum sampling methods and in 12 the area of high precision temporal measurement of voltage, which is the important process for 13 building high-fidelity devices. 14


Introduction
Quantum phenomena arising in optical current can be utilized in quantum com- 18 putation and quantum communication, specifically basing on those phenomena. There 19 are many optical-based implementation of quantum computation task, e.g.: the KLM 20 protocol proposed by Kneel et al. in [1], Photonic [2] or cat-qubits by Mirrahimi [3]. 21 In this paper we propose a new concept of implementation quantum computation in 22 optical current utilizing the distortion of homogeneous probability distribution over 23 photo detectors, caused by phase shifts between rails. We show how to implement in 24 optical current one component of Cosine series sampled operator (QCoSamp), that was 25 described in our previous work: [4]. 26 It is important, for all methods, the issue of measurement of the voltage caused by 27 photo-diode sensors, that we use as the terminal of optical current. It can work in one 28 of two modes: photovoltaic or photoconductive, however, due to necessity of, usually 29 weak, signal amplification, at the measurement point, we obtain the voltage, and it is at of light that intersect in two points, due to properly shaped geometry of the crystal. 49 Historically, we name one of such a point "signal" and the second one "idler", however 50 those two rays are nowadays combined to obtain greater beam intensity. Hence in the 51 resulting ray, there appears time-correlated (which mean indistinguishable) photon pairs 52 with cross -correlation. They are superposed as well, certainly. Therefore, after going 53 through the beam splitter, we obtain two cross-polarized rays. Because the photons are 54 superposed and time indistinguishable, they are entangled, which manifests itself in 55 measuring the time correlation of set of photons (see Magnitskiy et al. in [6]). It was 56 proven experimentally for ten photons entanglement by Wang et al. in [7] or for twelve 57 by Zhong et al. in [8]. The prove lays on the detection of time coincidence of photons 58 obtained in the photon detector. The laser beam is very narrow down to 10 fs with up to 59 20M beams per second. The measurement is made by photon counters ending each ray, 60 and the Correlation Counter, which returns the number of photons which appears in both 61 detectors in the same time interval of the 81ps length.

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In this context we communicate the idea how to implement the QCoSamp compo-63 nent. It was described in our previous work [4], leads to obtain the function 1 2 (1 − cos(x)).

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In this work, the changes of the variable x are achieved by changing the phase in phase 65 gate P φ . Hence, the idea is to prepare the quantum state in such a way that with changing 66 the parameter φ we obtain the desired value. If we dispose such a component we can 67 add a number of them to generate the series. We proved (ibidem) that such a series 68 maps into Fourier series. Therefore using it together with other quantum techniques, 69 like distributed phase encoding (Ruiz-Perez et al. in [9]) and amplitude amplification 70 (see [10,11]) we can create a number of algorithms, e.g., curve fitting, and even signal 71 and image extraction.

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In this section we present the quantum description of optical phenomena used in 74 quantum computation. Then we present selected protocols of quantum computation.

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Finally we present the optical setup that was used in our experiments. The quantum mechanical description of optical elements is based on two concepts.

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Firstly, we will use the Fock space, which describes the quantum state made of many pho-79 tons. Consider, that eigen basis of the photons consists of k eigen states (e.g. polarization).

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The basis state of the whole system of n photons is the sequence of numbers ϕ j being 81 cardinalities of photons which are in the state j. The states are called modes; the state 82 |0⟩ = |0...0⟩ is called vacuum. Secondly we define the annihilationâ and creationâ † opera-83 tors as follows:â|0⟩ = 0,â|n⟩ = √ n|n − 1⟩,â † |n⟩ = √ n + 1|n + 1⟩. The Lie algebra of 84 such operators is given by commutators: With such tools we can define optical devices (after Kok et al. [12]) by defining the 86 number of inputs and outputs, set of parameters, and evolution operator acting on the 87 photons going through the device. Those operators will be defined as a set of creation 88 operators for each output separately, depending on parameters and creation operators 89 for inputs.

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The simplest optical device is the single-mode phase shifter (PS) changing the phase 91 of incoming photons by phase φ that has one input S i and one output S o . Its evolution 92 operator is defined as follows: Beam splitter (BS) or Polarization rotator has two inputs B i1 , B i2 , two outputs B o1 , B o2 94 and is parametrized by R = cos 2 θ reflection rate and T = 1 − R = sin 2 θ transmission 95 rate. Furthermore we assume that the incoming waves can differ by the phase φ. In that 96 case: Qubit is represented by element of SU (2). The photon-based implementation of 98 such a group can be made using the photon polarization creating polarization qubits. It one qubit is defined as follows: Let's assume, that on the input of beam splitter, there appears n photons to the o1 input and m photons to the o2 input. We can formally denote it with annihilation and creation operator in Fock space: |n, After coming through the beam splitter this input operators will change to the output operator that transforms according to the eq. 2. Using this transformation (and Pythagorean identity) we can write the input operator in therms of output operators, as follows: Hence, the output state of the beam splitter for an input |n, m⟩ F is equal to:

KLM protocol 107
Let's consider the system of two qubits, one polarized vertically and one horizon-108 tally and let them pass through the beam splitter. Using creation operator notation, on 109 the output of the beam splitter we obtain both output operators acting on the vacuum: If we consider BS with R = T = 1/2, (hence θ = π/4) and phase φ = π/2 then sin(2θ) = 1, cos(2θ) = 0 and e iπ/2 = −1, e −iπ/2 = 1 and the system is described as follows: The above equation means the absence of the output state |1, 1⟩, which means that after

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In our previous work, we defined theoretically the QCoSamp method of quantum computation [4], basing on the quantum sampling, which creates the probability distribution over the measurement basis. The results are interpreted in the context of trigonometric series made of components: 1 2 (1 + cos(nx + r)). The light creates a probability distribution over the detectors, that could be recognized as the measurement basis of the optical quantum system. On the other hand, the state of the photon on the output of the BBO crystal in SPDC phenomenon is equal to (after Magnitskiy et al., ibidem, pg. 620): |HV⟩ + e iδ |V H⟩ √ 2 (8) . Then they goes through the 50%/50% beam splitters (pairs from each BBO separately). The circles outside the area of MOS symbolize its 4 output rails. Because of Hong-Ou-Mandel effect they groups in one of beam splitter outputs, so on the output we obtain |20⟩ or |02⟩ states. The final state is the tensor product of those two, so we obtain one of 4 states |2000⟩, |0200⟩, |0020⟩, |0002⟩. Nevertheless, this description of state is not sufficient, since the cross-Kerr non-linearities generates the entanglement in the polarization domain, which is not covered in this description. Therefore, we propose the notation of state visualized on the right side of figure above, where for each mode there are two slots in the ket: first for horizontal polarization and second for vertical one. This notation will be called rail-polarization (RP) notation.
Wang et al. [7] and Zhong et al. [8] uses SPDC with more sophisticated nonlinear units for generation of 10 and 12 photons. They uses the polarization beam splitter to obtain so called Greenberger-Horne-Zeilinger (GHZ) state for n photons: The technique (see fig. 1 involves to create n 2 pairs of photons using nonlinear unit (e.g.,

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BBO) in series. Then the entanglement is propagated by n 2 polarizing beam splitters.

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Each of them connects the existing entanglement with the next pair generated in the 165 line. In our work, we use similar setup, but for computation we need more states then 166 just GHZ, therefore, we connect both sides of the outputs from non-linear unit and we 167 use non-polarizing beam splitter, which creates together very interesting states, from 168 quantum computation perspective. Let's apply the 50:50 beam splitter on this state for obtaining the two building blocks of the optical setup (see fig. 2). Since the input state is |1, 1⟩ F , we will use eq. 4. The parameter θ is still equal to π/4, so cos(2θ) = 0, sin(2θ) = 1. We assume here for simplicity that the phase δ is equal to zero, nonetheless we have to remember, that without any involvement, this will not be a case. how to achieve it will be described later during defining the setup calibration process. The Magnitskiy equation 8 in the rail-polarization (RP) notation (see right part of fig. 2) will be denoted 3 : Moreover, we will denote the creation operators for the first input of a beam splitter with  fig. 3.

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Here we will provide a list of two and four creation operators, that will be useful in the remaining part of this work. Right now we need just two of them (namely: , but it is convenient to have them in one place. We don't provide proofs because one can easily derive them using the fact that creation operators commutes each other and expand the formula 5. We assume that the BS is 50%-50% and on the one output it has the phase shifter with the relative phase φ. The list is as follows:

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To have all our puzzles together, we create the list describing the transformations of states needed in our computation model, twin to aforementioned list of operators, as follows: In the virtue of above, the acting of BS (50%-50%, φ = 0) on the state on the output of BBO crystal is as follows: The above equation means, that while two entangled photons just after leaving the 174 nonlinear unit are not in the same rail, after crossing in beam splitter are both in "left" or 175 "right' rail. Nevertheless the original state (eq. 8) doesn't change at all.

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MSO (see fig. 2) consists of two such operations, therefore at the output there are four rails and four photons, entangled in pairs, for now. Therefore the final output state |MSO⟩ is a tensor product of two states |Ψ 1 ⟩ for the first nonlinear unit and |Ψ 2 ⟩ for the second one. Therefore, we can write the final state of MSO as follows: |11, 00; 11, 00⟩ + |11, 00; 0011⟩ + |00, 11; 11, 00⟩ + |00, 11; 00, 11⟩ We have so far assumed that the relative phases of the light in both nonlinear The calibration has to be done every time the setup will be changed and after a long 194 period of time since the last one. four output rails. Its state is given by eq. 14. The rails: one and two contains entangled color on the scheme) and two outboard pairs.

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there are 8 such states they will take 1 2 of the probability. We can omit them in our 230 considerations and treat as a noise.

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The last question in out computation model is, how, in fact, is a qubit represented in

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In above representation we can observe just first and second ququartit. We sum-up 243 the probability distribution obtained for this ququartits for the basis states: |01⟩, |02⟩ for 244 the parameter φ and |31⟩, |32⟩ for the parameter γ. 245 Now, if we apply the value of x to the top -left beam splitter, 2x to the top-right, the value r to the bottom -left and s to the bottom right (see fig. 4, and the resulting probability distributions will be multiplied by 4 we obtain the final function in the form: This function is the realization of QCoSamp made of two components.

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To conclude the method we have to indicate what part of the whole current is a 247 single QCoSamp Component and the method of addition of two or more component. from the output state of MOS, which gives 1 16 normalization factor for probability. It was omitted in this considerations for simplicity. 5 ditto 6 Three dimension qudits are called: qutrits Using more non-linear units and crossing theirs output as was described above, one can 254 construct the setups for more components.

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In current work we showed how to build an optical setup to create single component 257 for QCoSamp operator and how to add two or more components to create a series 258 described in our previous work [4]. Nonetheless the described setup offers more forms The detection of temporal correlation of 4 and more photons in two polarization states.

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For the purpose of our setup there is a need of specific detection that will point the 267 ququartits states. So for example the state |00⟩ has to be detected by observing the 268 remaining part of the system. On the other hand the states with two photon on the 269 same rail with the same polarization (e.g, |20⟩) has to be rejected, since we don't 270 use it for computations.

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• The implementation of amplitude amplification algorithm [10,11], which the core 275 of optimization methods that can be applied in QCoSamp.