Quantum Computing Simulation Of Quantum Interrogation
Quantum Computing Simulation of Quantum Interrogationby Shiloh Andersson, College of San MateoMentor: Alexander WongAbstractQuantum interrogation is the technique for measuring a quantum system without interacting with ordisturbing the system. In 1995, Paul Kwiat, along with a group of physicists, devised a method forquantum interrogation that allows for successful interaction-free measurements, and the success ratedepends on the number of times, N, a photon passes through a beam splitter. For the case of N 3,the success rate is about 42%. Using software provided by IBM, we design and implement a quantumcomputing algorithm that performs the Kwiat’s method for quantum interrogation, and we perform a ttest to assess the validity of the hypothesized success rate.1 IntroductionImagine two people, Alice and Bob. Alice presents Bob with a sealed box containing a photonsensitive bomb, and she asks him to determine whether or not the bomb will explode. If Bob answerscorrectly, Alice will reward him with several bars of gold. However, if Bob opens the box and thebomb is alive, the bomb will explode and Bob will die. Is there a way for Bob to win the gold withoutrisking his life? Physics says yes.The problem above can be solved with quantum interrogation, a technique for measuring quantumsystems without disturbing the system. In this case, Bob measures the state of the bomb withoutopening the box and exposing the bomb to light. There exists a method for quantum interrogation,proposed by Avshalom Elitzur and Lev Vaidman (Tan-Holmes, 2016). With this method, there is a25% chance that Bob can successfully determine the state of the bomb without detonating the bomb.But can we achieve a success rate that is higher than 25%?In 1995, researchers at the University of Illinois proposed a new method that builds upon the ElitzurVaidman scheme (Kwiat et al., 1995). For the case of N 3, where N denotes the number of times aphoton passes through a beam splitter, the Kwiat et. al scheme predicts that the success ratebecomes 42.1875%.In early 2019, researchers at the Indian Institute of Science Education and Research Kolkata usedIBM’s quantum computing software, QISKit, in order to create a quantum computing simulation of theElitzur-Vaidman scheme. We will extend their work to design and implement a quantum computingalgorithm for a simulation of the Kwiat et al. scheme, and we will use statistical testing to assess thehypothesized 42.1875% success rate.2 General Background2.1 Quantum InterrogationQuantum interrogation is the technique for measuring a quantum system (that is, a system composed of atoms or subatomic particles) without disturbing or interacting with the system. In essence,one takes a measurement on a quantum system, but there is no interaction with the system. For thisreason, quantum interrogation is also called “interaction-free measurement.” Naturally, one maywonder what constitutes as a measurement. For this paper, we will define “measurement” using the Think You?! The Proceedings of the Bay Honors Research Symposium, 2020. All Rights Reserved.
Andersson 2definition given by Mithuna Yoganathan, a PhD student studying quantum computing at the Universityof Cambridge, in her video on the difference between measurement and interaction. According toYogonathan, a measurement is, in essence, an action that provides information about a system to anobserver outside of the system (Yoganathan, 2018). As for the definition of “interaction,” we will takean interaction to be a bomb explosion (Raj et al, 2019). Thus, an interaction-free measurement forBob would be determination of the state of the existence of a bomb in the box without causing anexplosion.2.2 A Brief Introduction to Quantum MechanicsQuantum interrogation is possible due to key principles in quantum mechanics, the branch of physicsthat is concerned with atoms and subatomic particles. A key feature of quantum mechanics is waveparticle duality: quantum objects can behave both like waves1 and particles. One consequence of thewave-like nature of particles is quantum superposition, the idea that particles can exist in multiplestates at once prior to measurement. Before we examine the concept of quantum superposition, wewill first discuss wave superposition in classical physics.In classical physics, the principle of wave superposition states that when two waves overlap, theirwave functions add together to form a new wave. The function that represents each wave is amultivariable function in terms of position x and time t. If we had two waves, which are modeled bythe functions f0(x, t) and f1(x, t), then the resulting wave f(x, t) would bef 𝑥, 𝑡𝑓 𝑥, 𝑡𝑓 𝑥, 𝑡 .(2.1)In general, when we have n waves overlapping, the resulting wave due to superposition would bef 𝑥, 𝑡𝑓 𝑥, 𝑡 ,(2.2)where fi(x,t) represents each of the n waves. We will see a very similar equation when we discuss thesuperposition of quantum states and the relevant mathematics for quantum mechanics.Now we can transition to quantum physics. Quantum physics says that the state of a quantum systemor object, much like classical waves, can experience superposition before measurement of the state.That is, if the state of a quantum system remains unknown to an observer outside the system, thenthe system exists in all possible states at once; when an outside observer attempts to measure thesystem, then quantum superposition is broken and the system collapses to a single state. Let’s lookat quantum superposition through an example with a photon and beam splitter (a device that reflectsa portion of a beam of light and allows the rest of the light to transmit through the beam splitter).Suppose there is a closed system containing a photon and a beam splitter. The photon is shot at thebeam splitter, and it has an equal probability of taking one of two paths, horizontal⟩2 and vertical⟩.1It should be noted that quantum particles do not behave like literal waves; rather, these particles can be mathematicallymodeled by wave functions (Yoganathan, 2019).2Notation for a quantum state.
Andersson 3(Figure 1). Classical physics would predict that prior to measurement, the photon is either in one stateor the other. However, because quantum states behave like waves, these states can experiencesuperposition. As long as the photon’s path is unknown to any object outside the system, then thephoton is in a superposition of both horizontal⟩ and vertical⟩.The photon’s ability to exist in asuperposition of both states is the mechanism that allows quantum interrogation to occur, as we willsee in the quantum interrogation technique proposed by Elitzur and Vaidman (Tan-Holmes, 2016).Figure 1: An example of quantum superposition. The photon is aimed horizontally at the beam splitter, and once it passes through thebeam splitter, it is in a superposition of traveling vertically and horizontally, so long as the photon’s path is indeterminate.Similar to classical waves, we can represent a superposition state as a linear combination of quantumstates, which we will further explore in Section 2.3. For now, we will discuss the implications ofsuperposition in both classical and quantum mechanics.The property of superposition is vital to physics because this principle allows for wave interference. Inclassical physics, waves can constructively and destructively interfere with each other (see Figure 2).Constructive interference occurs when the peaks of two waves align, allowing the waves to “add up”so that the resulting wave has an amplitude that is the sum of each of the two waves’ amplitude.Destructive interference occurs when the peaks of one wave align with the troughs of another wave,making the two waves “cancel out.”Figure 2: The two waves on the left demonstrate constructive interference. The two waves on the right demonstrate destructiveinterference.
Andersson 4Extending the idea from classical mechanics to quantum mechanics, the wave properties of quantumobjects allow a quantum object to interfere with itself. Interference effects allow us to determine thestate of the system which we wish to measure, which is, in our case, the state of the bomb. We willexplore the importance of superposition in Section 2.4.2.3 Mathematics for Quantum MechanicsLet us imagine a quantum system whose state Ψ⟩ is the superposition of Ψ ⟩ and Ψ ⟩. Much likeclassical waves, we can express the superposition state Ψ⟩ as a linear combination of Ψ ⟩ and Ψ ⟩,giving us: Ψ⟩α Ψ ⟩β Ψ ⟩The coefficients in front of each state is related to the probability of measuring the state. Tocalculate the probabilities, we square the absolute value of the coefficients3:P Ψ ⟩ α 𝑃 Ψ ⟩ 𝛽 Furthermore, the sum of the probabilities must add up to 1, so α β 1.Example:There is some system whose state is Ψ⟩𝑖 21 1⟩2 0⟩The probabilities of measuring each state areP 0⟩𝑃 1⟩𝑖 21 2121,2and these probabilities sum up to 1, so the coefficients for each state are valid.Now let us generalize for a quantum system that is in a superposition of n states, where n is somepositive integer. The state Ψ⟩ is Ψ⟩ρ Ψ ⟩ ,(2.3)where ρi is the coefficient for each state Ψ ⟩. The resulting probability of some state Ψ ⟩ is𝑃 Ψ ⟩ ρ ,3Sometimes, coefficients contain negative or complex values, so we must take the square modulus of the coefficient inorder to avoid the chance of negative probabilities.
Andersson 5(2.4)and the sum of the probabilities is 𝜌 1.(2.5)Notice that Equation (2.3) resembles Equation (2.2). This is unsurprising because while one equationrepresents quantum states and the other represents classical waves, both equations describe thephenomena of superposition.Now that we have discussed the necessary mathematics and key concepts of quantum mechanics,let us see how these ideas permit quantum interrogation.2.4 The Elitzur-Vaidman SchemeThere are several ways to perform quantum interrogation. A simple technique for quantuminterrogation was developed by Avshalom Elitzur and Lev Vaidman (Tan-Holmes, 2016). The setupfor their technique (pictured in Figure 3) consists of two beam splitters, two mirrors, two detectors, thebomb, and a photon.Figure 3: General setup for the Elitzur-Vaidman scheme. The blue rectangles represent the beam splitters; the grey rectanglesrepresent the mirrors; A and B denote the detectors; the black rectangle is the bomb; the black circle is our photon.To understand how their setup permits quantum interrogation, we will begin by imagining two cases:there is a dead bomb, and there is a live bomb.Case 1: Dead BombOur photon will enter the setup horizontally. Then it will approach the beam splitter. The photon canmove vertically, or it can move horizontally and approach the bomb. Because the bomb is dead, thephoton will not detonate the bomb; the photon will move through the bomb and toward the mirror onthe bottom (see Figure 4). If the bomb is dead and the photon moves horizontally, the photon willmove toward the mirror. Likewise, if the photon moves vertically, then it will move toward the mirroron the top (see Figure 4). In either case, once the photon reflects off either mirror, it will approach thesecond beam splitter and hit a detector.
Andersson 6In such a setup, there is no way of figuring out whether the photon had initially moved horizontally orvertically. Though the detectors indicate the presence of a photon, they do not give any informationabout the photon’s path. Therefore, quantum superposition and interference are permitted in such acase. When the photon passes through the first beam splitter, the photon will then exist in asuperposition of moving horizontally and vertically. As it reflects off the mirrors and approaches thesecond beam splitter, the photon will then interfere with itself. It constructively interferes to movetoward detector B and destructively interferes toward detector A, so the photon always appear ondetector B.Figure 4: Case 1 of the Elitzur-Vaidman Scheme: The bomb is dead. The photon enters horizontally, and quantum superposition andinterference cause the photon to always appear on detector B.Case 2: Live BombUnlike the first case, superposition is not permitted, and interference will not occur. In this case, if thephoton continues to move horizontally, then it will certainly cause the bomb to explode. A bombexplosion guarantees that the photon moved horizontally, giving the observer information about itspath. So if the photon moves horizontally, a bomb explosion occurs, and the photon is “scattered” andnever appears on either detector (Elitzur & Vaidman, 1993).If, however, the photon is reflected off the beam splitter, it will then move vertically and hit the mirror,and them it has an equal chance of hitting either detector. Because of the reflectivity of the beamsplitters, there is a 50% chance the bomb explodes, and a 50% chance it does not explode. In thecase that it does not explode, the photon is equally likely to hit either detector, which means a 25%chance of hitting detector A and a 25% chance of hitting detector B.Figure 5: Case 2 of the Elitzur-Vaidman Scheme: The bomb is alive. The photon still enters horizontally, but superposition andinterference no longer occur; the photon will either detonate the bomb, or it will have an equal probability of showing up on eitherdetector.
Andersson 7In summation, if the bomb is dead, it will always appear on detector B. If the bomb is alive, then it hasa 50% chance of exploding the bomb (failure), a 25% chance of appearing on detector B (neithersuccess nor failure, as bomb could be dead or alive if photon appears on B), and a 25% chance ofappearing on detector A (success).2.5 The Kwiat et al. SchemePaul Kwiat and a group of physicists devised a new method for quantum interrogation, which buildsupon the EV scheme. If we look back at the EV scheme, we see that the steps in their method are:1.2.3.4.5.Pass through beam splitter.Possibly detonate the bomb.If the bomb has not exploded: Reflect off the mirror.Pass through second beam splitter.Measure state of photon to determine state of bomb.The Kwiat et al. scheme is quite similar, but there are a few modifications. While this method still usesa photon, bomb, and two mirrors, the Kwiat et al. scheme actually only uses one beam splitter. Thesetup for this method appears as follows:Figure 6: Kwiat et al. scheme setup: A photon is placed in the left side of the cavity. The bomb lies in the right side.Instead of using multiple beam splitters, this method simply allows the photon to pass through thesame beam splitter over and over again. Let us examine the path of the photon when we let thephoton pass through 3 times. That is, when N 3. (Note: The following figures depicts the timeevolution of the photon traveling through the Kwiat et al. setup, not multiple bombs/beamsplitters/pairs of mirrors.)Figure 7: When the bomb is dead. Photon always ends “up” (actually ends right, but this image has been rotated 90º).
Andersson 8Figure 8: When the bomb is alive. If the photon ends “down” (actually ends left, but this image has been rotated 90º), we know thebomb is alive without detonating it; otherwise, the photon is scattered and will not be measured.Note: If the photon hits the bomb, that is the end of its path. Figure 8 merely portrays all possiblepaths.Based on the above figures, we find that the steps for the Kwiat et al. scheme are as follows:1.2.3.4.Pass through beam splitter.Possibly detonate the bomb.If the bomb has not exploded and if this cycle is not the Nth cycle: Reflect off the mirror.Repeat steps 1-3 until the photon passes through the beam splitter N times/until the bombexplodes.5. Measure state of photon to determine state of bomb.In order for the Kwiat et al. scheme to work, the reflectivity of the beam splitter (that is, the proportionof light the beam splitter reflects or the probability a photon is reflected) has to be adjustedaccordingly for N cycles. Furthermore, the success rate will be adjusted based on the reflectivityaccording to these equations:π𝑅 ,(2.7)where R is the reflectivity and P is the success rate (Kwiat et al., 1995). For the case of N 3, we findthat we have a reflectivity of:𝑅𝑐𝑜𝑠𝜋2 30.75(2.8)𝑃(2.9)𝑐𝑜𝑠𝜋2 30.750.42187542.1875 %
Andersson 9But this 42.1875% is only a hypothesized value, proposed by Kwiat et al. If one were to perform anexperiment of the Kwiat et al. scheme for N 3, would they observe a 42.1875% success rate?2.6 Quantum Computing Simulation of the EV SchemeIn early 2019, researchers at the Indian Institute of Science Education and Research Kolkatadeveloped a quantum computing algorithm to verify the 25% success rate for the EV scheme (Raj etal., 2019). Like Raj et. al, we intend to use quantum computing to assess the validity of thehypothesized 42.1875% success rate by creating our own quantum computing simulation of the Kwiatet al. scheme.3 MethodologyUsing IBM’s quantum computing software QISKit (2019), we will extend the work done by Raj et al.by designing and implementing a quantum computing algorithm that will simulate the Kwiat et al.scheme for the case of N 3. Then we will run the simulation 40 times, where each simulation runsthe circuit 5000 times, in order to obtain the mean and standard deviation of the success rate. Usingthe mean and standard deviation, we will then perform a t-test in order to assess the hypothesis thatthe success rate is 42.1875% for N 3.3.1 Qubits for this SimulationOur quantum computing simulation will have three qubits: photon0, photon1, and bomb0. photon0represents the direction of our photon, where 0⟩ indicates that the photon moves to the left and 1⟩indicated that it moves to the right. photon1 represents whether the photon has “scattered” or not(Elitzur & Vaidman, 1993), where 0⟩ means the bomb did not explode and the photon has notscattered, and 1⟩ means that the bomb exploded and the photon has scattered. Finally, the qubitbomb0 indicates the presence of a bomb, where 0⟩ means that the bomb is dead and 1⟩ means thatthe bomb is alive. At the end of our circuit, we will measure the state of qubits photon0 and photon1 ina combined state photon0photon1⟩.Our photon will begin by moving to the right. If the bomb is dead, then we should measure thephoton’s direction to be toward the right, which is a state of 10⟩. However, if the bomb is alive, thenwhen the bomb passes through the beam splitter, it will either continue moving right and hit the bomb,or it will move left and not detonate the bomb. This means that we should expect two possible states: 00⟩ and 11⟩. Based on Equation (2.9), we should expect to see 00⟩ measured in about 42% of thesimulations.3.2 Quantum Gates Used in the SimulationListed below are the matrix forms of the quantum gates used to represent the components of oursetup for quantum.Beam splitterEquation (2.8) tells us that our beam splitter should have a 0.75 reflectivity. Thus, when light passesthrough our beam splitter, 75% of the light wave will be reflected and undergo a phase shift, while25% of the wave will pass through without any reflection or phase shift. However, we are interested ina single photon. When a photon passes through our beam splitter, we should observe a 75%probability of measuring the photon in the opposite state with a phase shift, and a 25% chance ofmeasuring the photon in its original state. The following matrix will represent such a beam splitter:
Andersson 10𝐵𝑆1 2 𝑖 3 2𝑖 32 1 2 Let us examine the effect of applying BS onto some quantum states.Examples of Applying Our Beam Splitter, BS, onto Qubits:1. BS on 0⟩𝐵𝑆 0⟩1 𝑖 3 22 1 𝑖 3 1 02 21 2 𝑖 3 2 1 12 0𝑖 3 02 1𝑖 3 1⟩21 0⟩2Squaring the modulus of the coefficients, we find that there is a 0.25 probability of 0⟩ and 0.75 probability of 1⟩. Since we started with 0⟩, these probabilities tell us that there is a 75% chance that we measure the qubitto be in the opposite state, which corresponds to a reflectivity of 0.75.2. BS on 1⟩𝐵𝑆 1⟩1 𝑖 3 22 0 𝑖 3 1 12 21 02 1𝑖 3 12 01 1⟩2𝑖 3 0⟩2Again, we square the modulus of the coefficients, and we find that there is a 0.75 probability of measuring 0⟩and 0.25 probability of 1⟩. Similar to example 1, we observe a 0.75 probability of measuring a qubit in theopposite state (in this case, the qubit is 1⟩ and the opposite state is 0⟩) after applying the beam splitter.Now that we have a matrix to model our beam splitter, we will need a quantum gate to represent sucha matrix so that we can utilize IBM’s quantum computing software, QISKit. QISKit has no matrix of theform BS, but QISKit has a U3 gate that can represent any matrix, depending on the parameters thatwe feed into the gate. The general U3 gate is given by:𝑼𝟑 𝛉, 𝛟, 𝛌𝒆𝒊𝝀 𝒔𝒊𝒏 𝛉/𝟐𝒆𝒊 𝛌 𝛟 𝒄𝒐𝒔 𝛉/𝟐𝒄𝒐𝒔 𝛉/𝟐𝒊𝛟𝒆 𝒔𝒊𝒏 𝛉/𝟐.If we pick θ, ϕ, and λ to be 2π/3, π/2, and π/2 for our U3 gate, we will obtain:𝑈 2π/3, π/2, π/2cos π/3𝑒 / sin π/3𝑒𝑒/sin π/3cos π/31 2 𝑖 3 2𝑖 32 .1 2 As can be seen, these values for the parameters generates a matrix that matches our beam splitterBS. From now on, we will refer to BS as U3.
Andersson 11MirrorsWhen light hits a mirror, the light wave is reflected, and the wave undergoes a phase change. Thefollowing matrix represents a mirror:0 𝑖,𝑖 0𝑀and we will explore the consequence of letting a photon hit the mirror.Examples of Applying the Mirror, M, onto Qubits:1. Applying the mirror on 0⟩0 𝑖 100𝑖𝑖 1⟩𝑖 0 0𝑖1When we apply the mirror on 0⟩, we see that the state flips, and the i term represents a phase shift.𝑀 0⟩2. Applying the mirror on 1⟩𝑖0 𝑖 01𝑖𝑖 0⟩0𝑖 0 10Much like the first case, we find that applying the mirror on 1⟩ flips the state and shifts its phase.𝑀 1⟩QISKit, once again, has no gate whose matrix representation is M. However, in order to avoidconfusion, with the beam splitter, we will not use the U3 gate to generate a matrix for the mirror effect.Instead, we will utilize different quantum gates and the associative property of matrix multiplication toobtain a matrix representation for the mirrors. We will use the following gates:𝑋011,𝑌00𝑖𝑖,𝑍01001We will apply the gates on a qubit in this order: Y, X, Z, and X, which mathematically looks like:𝑋𝑍𝑋𝑌 state⟩4 .Because of the associative property of matrix multiplication, we can multiply the first four matrices intoa single matrix𝑋𝑍𝑋𝑌0 11 0100101100𝑖𝑖00𝑖𝑖,0which is our mirror M.Bomb EffectWhen the bomb is dead (that is, when bomb0 measures 0⟩), then the photon is completely unaffectedby the presence of the bomb. However, in the case that the bomb is alive (bomb0 measure 1⟩) andthe photon moves to the right (photon0 measures 1⟩), then the photon will detonate the bomb, andthe photon will scatter (photon1 flips from 0⟩ to 1⟩).The ccx gate takes in three qubits as its parameters. If the first two parameters are both 1⟩, then it4In this mathematical statement, the closer a gate is to the qubit, the earlier the gate is applied onto the qubit.
Andersson 12will flip the third qubit, like from 0⟩ to 1⟩. For our bomb effect, we will apply the ccx gate with theparameters photon0, bomb0, and photon1 in that order.3.3 Quantum Computing AlgorithmThe code for our algorithm can be found in the Appendix. Appendix A contains the general algorithmfor the Kwiat et al. scheme. Appendix B contains the code we run to test a live bomb, and Appendix Ccontains the code we run to test a dead bomb.4 ResultsFigure 9: Results when the bomb is dead. The state 10⟩ is measured 100% of the time, which is expected.QuantityNumerical ValueAverage SuccessRate1Standard Deviation0Table 1: Results from Testing the Dead BombFigure 10: Results when the bomb is alive. The state 00⟩ is measured about 42.1% of the time, and the state 11⟩ is measured about57.9% of the time; these probabilities add up to 100%.
Andersson 13QuantityNumerical ValueAverage SuccessRate0.42077Standard Deviation0.008374264606059908Table 2: Results from Testing the Live Bomb5 AnalysisIn order to determine if the success rate of interaction-free measurements is 42.1875% based on thedata from the quantum computing simulation, we will use statistical hypothesis testing. Specifically,we will perform a t-test.Our null hypothesis, H0, is: The success rate of interaction-free measurements for the Kwiat et alscheme with three cycles is 42.1875%.Our alternative hypothesis, Ha, is: The success rate of interaction-free measurements for the Kwiat etal scheme with three cycles is less than 42.1875%.For our hypothesis test, we will choose an alpha value of α 0.1. We pick a high value for α in orderto reduce the probability of a type II error. A type II error in this case implies that we erroneously failedto reject the null hypothesis that the success rate is 42.1875%. Thus, the reality would be that thesuccess rate is lower, which, in the example in the introduction, would suggest that Bob has a lowerchance of survival than anticipated.In addition to choosing a high value for alpha, we also chose to run 40 simulations in order to obtain40 samples of the success rate. By increasing the sample size, we have further reduced theprobability of a type II error.Below is the data used to perform the t-test on a TI-84 PLUS CE calculator. The table also includesthe resulting p-value.QuantityNumerical alue0.20453Table 3: Data and Results from the t-testThe p-value of 0.20453 is much greater than the alpha value of α 0.1. This means that we fail toreject the null hypothesis that the success rate is 42.1875% (that is, we lack the evidence to disprovethe success rate of 42.1875% for N 3). More testing would need to be done (either a different form
Andersson 14of hypothesis testing or varying the parameters of the t-test) before we can generally “accept”5 thisvalue.As of now, we find that the 42.1875% success rate is not false. If we do accept this value throughfurther testing, one may realize that, going back to the problem in the introduction, Bob has about a58% chance of exploding the bomb. For N 3, Bob is more likely to die than survive if the bomb isalive. Can we improve the success rate so that it is at least 50%? Yes.Recall Equations (2.6) and (2.7), which give us equations for the reflectivity and success rate for Ncycles. Using those equations, we find that as we increase N, both the reflectivity and success rateincrease. In the table below, we find that increasing N to 4 or 5 cycles already gives us a greater than50% success rate. Increasing to N 10, we get an even higher success rate of about 78%. As we letN get larger, the success rate approaches 100%.Cycles Reflectivity 1097.55%78.05%5099.90%95.18%Table 4: Success Rates for Different Numbers of CyclesIn the future, we would like to modify the current algorithm for N 3 so that we can verify the successrate for a much higher value of N, showing that the success rate does indeed approach 100% for asufficiently large value of N.6 ApplicationsThough we have framed quantum interrogation around the problem of Bob determining the state of abomb without an explosion, quantum interrogation actually has some real-world applications. One ofthese applications include reducing sample damage in electron microscopy, a laboratory techniqueoften used in biology or physics research (Putnam & Yanik, 2009).7 DiscussionAs far as we’re aware, we have used IBM’s quantum computing software to create the first quantumcomputing algorithm to perform a simulation of the Kwiat et al. scheme for the case of N 3. Ourresults fail to disprove that the success rate is 42.1875% for N 3. Further testing would need to bedone before we can accept that this success rate is likely true. In addition to doing more verification ofthe success rate for N 3, future work would include modifying the current algorithm in order to5In science, we never prove a theory. Instead, we find enough evidence that allows us to agree that the theory is true.
Andersson 15assess the success rate for a sufficiently large value of N. In modifying the algorithm so that it worksfor some large value of N, we would like to show that the success rate does indeed approach 100%.AppendixA Code# Importing libraries, and configuring accountfrom qiskit import *from qiskit.compiler import transpile, assemble from qiskit.circuit import *from qiskit.tools.jupyter import *from qiskit.visualization import *from numpy import *from statistics import mean, stdevfrom scipy import stats# Loading your IBM Q account(s)provider IBMQ.load account()# Make a photon with two qubits# First qubit is direction (0 left, 1 right)# Second qubit indications bomb explosion (0 not yet exploded, 1 exploded)photon QuantumRegister(2, 'photon')# Make a bomb with one qubit# (0 dead bomb, 1 live bomb)bomb QuantumRegister(1, 'bomb')# Measures the state of the photon# First classical bit is the second photon qubit# Second classical bit is the first photon qubit# State is written with second classical bit first, then first classical bitstate ClassicalRegister(2, 'state')# Create a circuit with the three registers abovecircuit QuantumCircuit(photon, bomb, state)# Number of times the photon passes through the beam splittern 3# Number of simulations used to get empirical probabilities# for the successes and failuressims 5000simulator Aer.get backend('qasm simulator')"""Set up circuit so that the photon moves right and nothing has exploded. bomb alive: True if the bombis alive, False if dead"""def setup(bomb alive):# Reset everything to ircuit.reset(bomb[0])circuit.measure([photon[0], photon[1]], [state[1], state[0]])# Allow photon to start moving rightcircuit.x(photon[0])# Set state of the bomb to alive, if applicable
Andersson 16if (bomb alive):circuit.x(bomb[0])"""Mirror part of the circuit.So long as the bomb has not exploded, the photon will reflec
the success rate is about 42%. Using software provided by IBM, we design and implement a quantum computing algorithm that performs the Kwiat's method for quantum interrogation, and we perform a t-test to assess the validity of the hypothesized success rate. 1 Introduction Imagine two people, Alice and Bob.
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