Performance of IBM's open-access quantum computers on different quantum circuits Author: Alex Mart´ınez Miguel. Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Advisor: Artur Polls Mart´ı (Dated: January 15, 2020) Abstract: We use the different quantum computers which IBM made available on the cloud (IBM Q Experience) to implement different quantum circuits. First, we study the dynamics of a single spin interacting with a magnetic field using the main single-qubit gates, which allows us to obtain the evolution of the three components of the spin. Finally, we study the performance of the quantum computers by implementing two quantum algorithms: the dense coding protocol, which we use to study the connectivity between qubits, and the quantum Fourier transform, which we use to study the performance of three different quantum computers when the number of qubits is increased. I. INTRODUCTION tum algorithms to analyze the performance of different quantum prototypes. Finally, in Section V, conclusions In the last four decades the field of quantum compu- are presented. tation has not stop growing and attracting people from different backgrounds, such as physics, computer science, II. FUNDAMENTALS information theory and cryptography [1]. Nowadays, it is a very promising and exciting field. Quantum comput- ers have been presented as potential tools for solving real A. The qubit problems which are considered out of reach for classical computers. It is believed that they could contribute to Quantum computers use qubits, which generalize the the development of new breakthroughs in science, medi- classical bit. They are two-level systems, i.e. systems cations, machine learning methods or materials to make with one quantum property that can take two possible more efficient devices and structures, among others. One values. These two values allow us to label the basis states such example would be a quantum algorithm for factor- and are denoted as j0i and j1i. The main feature of the ing integers [2], which was developed by Peter Shor in qubits is the possibility to have superposition states. A 1994. qubit can be in an arbitrary combination of the two basis To that end, big companies like IBM or Google [3] have states been investing huge amounts of money to build the first j i = αj0i + βj1i; (1) full-operational quantum computer. There is more, in May 2016, IBM launched the IBM Q Experience [4], an being α and β complex numbers. Taking into account open online platform which gives the general public ac- the normalization condition h j i = jαj2 + jβj2 = 1, it cess to a set of prototype quantum processors. follows that a general state of a qubit can be expressed The main purpose of this work is to implement well- in terms of two real parameters (up to a global phase) known quantum algorithms in the framework set up by j i = cos (θ=2)j0i + sin (θ=2)eiφj1i: (2) IBM. On the one hand, a study regarding the interac- tion of a single spin with a magnetic field will be done. Therefore, the quantum state of a single qubit can be vi- On the other hand, an inspection of the performance sualized as a vector of length 1 inside the so-called Bloch of these processors will be carried out by going through sphere. The parameters θ and φ are the polar and az- two specific quantum algorithms: superdense coding and imuthal angle of the spherical coordinates. quantum Fourier transform. From the set of quantum In the IBM Q Experience, the prototypes use a physical processors we will use three of them: two 5-qubit proto- type of qubit called superconducting transmon qubits, types, ibmqx2 and ibmq vigo, and the 15-qubit prototype which are made of superconducting materials. The two ibm 16 melbourne. states j0i and j1i represent two possible energy levels The study is organised as follows. In Section II, the con- within this superconducting system. For these systems cept of qubit will be explained and a brief description of to behave as an abstract qubit they must be at drastically the most relevant (to this work) quantum gates will be low temperatures (15 mK in the case of IBM's). made. In Section III, we study the interaction of a spin with a magnetic field. In Section IV, we use two quan- B. Quantum gates Quantum gates are unitary matrices (they must be re- ∗Electronic address: [email protected] versible and conserve the probability amplitudes) which Performance of IBM's open-access quantum computers on different quantum circuits Alex Mart´ınez allow to change the state of the qubit, e.g. create super- Si = σi~=2, the Hamiltonian describing this interaction positions, rotations, etc. is The most basic quantum gates are the single-qubit gates ! X,Y and Z (the usual Pauli matrices). In matrix form H = −~µ · B~ = ~ 0 σ ; (8) 2 z 0 1 0 −i 1 0 X = ;Y = ;Z = : (3) where ! = −gµ B = is the Larmor frequency. Since 1 0 i 0 0 −1 0 B 0 ~ we are dealing with a time-independent Hamiltonian, the solution of the Schr¨odingerequation for this system Two other important gates are the Hadamard gate, H, might be found applying the time evolution operator [5] which generates superposition, and the S gate, which U(t; 0) = exp (−iHt= ) to the initial state j (0)i adds a phase. In matrix form ~ j (t)i = exp (−iHt= )j (0)i: (9) 1 1 1 1 0 ~ H = p ;S = : (4) 2 1 −1 0 i In matrix form, the operator is written as The Hadamard gate acts as follows: Hj0i ≡ j+i = (j0i + p p !0t 1 0 U(t) = exp −i σ = : (10) j1i)= 2 and Hj1i ≡ |−i = (j0i − j1i)= 2. Applying the 2 z 0 ei!0t the Spgate to these new states: Sj+i ≡ j pi = (j0i + ij1i)= 2 and S|−i ≡ j i = (j0i − ij1i)= 2. We are We choose our initial state to be in a superposition of able to define three different bases: the computational both basis states basis: fj0i; j1ig, the x-basis: fj+i; |−i} and the y-basis: θ θ fj i; j ig. In Section III we will see how to make j (0)i = cos j0i + sin j1i: (11) measurements in each one of the bases. 2 2 To end with the single-qubit gates, we must introduce the U1, U2 and U3. In matrix form Thus, the evolved state will be given by iλ θ θ cos (θ=2) −e sin (θ=2) i!0t U (θ; φ, λ) = ; (5) j (t)i = cos j0i + e sin j1i: (12) 3 eiφ sin (θ=2) eiλ+iφ cos (θ=2) 2 2 Now, by computing the different components of the spin U1(λ) = U3(0; 0; λ);U2(φ, λ) = U3(π=2; φ, λ): (6) we get All the aforementioned gates are just special cases of ~ Mz(t) = h (t)jSzj (t)i = cos (θ); (13) these. 2 ~ Mx(t) = h (t)jSxj (t)i = sin (θ) cos (!0t); (14) Now lets turn to the the two-qubits quantum gates. 2 The most important one is the Controlled-NOT (CNOT) M (t) = h (t)jS j (t)i = ~ sin (θ) sin (! t): (15) gate. Its action is to flip the target qubit if the control y y 2 0 qubit is j1i; otherwise it does nothing. The last quan- tum gate useful for this work is the controlled phase shift Next, we are ready to implement the problem in a quan- tum circuit. First, we need to prepare the initial state. gate Rm, which is specially relevant for implementing the quantum Fourier transform. In matrix form they read All circuits in the IBM Q Experience are initialized in the j0i state, so, from this state we can obtain j (0)i by 01 0 0 01 01 0 0 0 1 applying the following unitary operation 0 1 0 0 B0 1 0 0 C CNOT = B C ;Rm = B C : (7) cos (θ=2) − sin (θ=2) @0 0 0 1A @0 0 1 0 A Uin = ; (16) 2πi sin (θ=2) cos (θ=2) 0 0 1 0 0 0 0 e 2m which is obtained by using the U3(θ; '; λ) quantum gate and setting ' = 0 and λ = 0. The next step is to im- III. SINGLE-SPIN DYNAMICS plement the time evolution operator, which is given by Eq. (10). This is accomplished by using the U1(λ) if we Firstly, we study the interaction of a single spin with identify λ = !0t. Finally, we must make a measurement. a time independent magnetic field along the z-direction IBM Q Experience only allows to make a z-measurement ~ B = B0z^. The objective is to analyze how to build the directly, which is the one we must use to compute Mz(t). quantum circuits that allow us to compute the three com- To find the x-component of the spin, we must measure ponents of the spin as a function of time. in the x-basis, and this is done using circuit (b) of Fig. 1. ~ Given the magnetic moment of the spin ~µ = gµBS=~, Same for the y-component, the measurement in the y- which is expressed in terms of the Pauli matrices σi from basis is shown in circuit (c) of Fig. 1. Treball de Fi de Grau 2 Barcelona, January 2020 Performance of IBM's open-access quantum computers on different quantum circuits Alex Mart´ınez j0i H Z H j0i U3 U1 (a) j0i j0i U3 U1 H (b) y j0i U3 U1 S H (c) FIG.