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Quantum Simulation of the Single-Particle Schrodinger Equation

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Quantum Simulation of the Single-Particle Schrodinger Equation Quantum simulation of the single-particle Schr¨odinger equation 1,2, 3, Giuliano Benenti ∗ and Giuliano Strini † 1CNISM, CNR-INFM & Center for Nonlinear and Complex Systems, Universit`adegli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy 2Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy 3Dipartimento di Fisica, Universit`adegli Studi di Milano, Via Celoria 16, 20133 Milano, Italy (Dated: May 21, 2018) The working of a quantum computer is described in the concrete example of a quantum simulator of the single-particle Schr¨odinger equation. We show that a register of 6 − 10 qubits is sufficient to realize a useful quantum simulator capable of solving in an efficient way standard quantum mechanical problems. I. INTRODUCTION II. QUANTUM LOGIC 1,2 The elementary unit of quantum information is the During the last years quantum information has be- qubit (the quantum counterpart of the classical bit) and come one of the new hot topic field in physics, with the a quantum computer may be viewed as a many-qubit potential to revolutionize many areas of science and tech- system. Physically, a qubit is a two-level system, like nology. Quantum information replaces the laws of classi- 1 the two spin states of a spin- 2 particle, the polarization cal physics, applied to computation and communication, states of a single photon or two states of an atom. with the more fundamental laws of quantum mechanics. A classical bit is a system that can exist in two dis- This becomes increasingly important due to technological tinct states, which are used to represent 0 and 1, that progress reaching smaller and smaller scales. Extrapolat- is, a single binary digit. The only possible operations ing the miniaturization process of integrated circuits one (classical gates) in such a system are the identity (0 0, would estimate that we shall reach the atomic size for 1 1) and NOT (0 1, 1 0). In contrast, a quantum→ storing a single bit of information around the year 2020 bit→ (qubit) is a two-level→ quantum→ system, described by (at the present time the typical size of circuit compo- a two-dimensional complex Hilbert space. In this space, nents is of the order of 100 nanometers). At that point, one may choose a pair of normalized and mutually or- quantum effects will become unavoidably dominant. thogonal quantum states, called 0 and 1 , to represent | i | i The final aim of quantum computation is to build a the values 0 and 1 of a classicalbit. These two states form machine based on quantum logic, that is, a machine that a computational basis. From the superposition principle, can process the information and perform logic operations any state of the qubit may be written as in agreement with the laws of quantum mechanics. In ψ = α 0 + β 1 , (1) addition to its fundamental interest, a large scale quan- | i | i | i tum computer, if constructed, would advance computing where the amplitudes α and β are complex numbers, con- power beyond the capabilities of classical computation. strained by the normalization condition α 2 + β 2 = 1. Moreover, state vectors are defined up to| a| global| | phase The purpose of the present paper is to illustrate the of no physical significance. Therefore, ψ only de- main features of a quantum computer in the concrete pends on two real parameters and we may| writei ψ = example of a quantum simulator of the single-particle cos θ 0 + eiφ sin θ 1 , with 0 θ π and 0 φ<|2πi. Schr¨odinger equation. We wish to show that quantum 2 | i 2 | i ≤ ≤ ≤ arXiv:0709.1704v2 [quant-ph] 30 Nov 2007 A collection of n qubits is known as a quantum register logic, already with a small number of qubits, allows the of size n. Its wave function resides in a 2n-dimensional efficient simulation of basic quantum mechanical prob- complex Hilbert space. While the state of an n-bit classi- lems. A quantum simulator with 6 10 qubits could − cal computer is described in binary notation by an integer produce a real picture book of quantum mechanics. We k 0, 1, ..., 2n 1 , believe that snapshots from this book will help develop a ∈{ − } n 1 physical intuition of the power of quantum computation. k = kn 1 2 − + + k1 2+ k0 , (2) − ··· Our paper adds to other existing pedagogical presenta- with k0, k1,...,kn 1 0, 1 binary digits, the state of − ∈ { } tions of various apects of quantum computation, includ- an n-qubit quantum computer is 3 ing Shor’s quantum algorithm for integer factorization , 2n 1 4,5 − Grover’s quantum search algorithm , quantum infor- ψ = c k , (3) | i X k | i mation processing by means of nuclear magnetic reso- k=0 nance techniques6, using linear optics7, with cold atoms 8 9 where k kn 1 k1 k0 , with kj state of the j-th and ions , or with superconducting circuits , quan- n− 10 11 | i ≡ | 2 1i · · ·2 | i| i | i tum measurements , decoherence , quantum error- qubit, and k=0− ck = 1. Note that we use the short- 12 P | | correction and fault-tolerant quantum computation . hand notation kn 1 k1 k0 for the tensor product | − i · · · | i| i 2 kn 1 k1 k0 . Taking into account the nor- sources (energy) grows only linearly with n. malization| − i ⊗ · · condition· ⊗ | i ⊗ and| i the fact that a quantum state To implement a quantum computation, we must be is only defined up to an overall phase, the state of an able to control the evolution in time of the many-qubit n-qubit quantum computer is determined by 2 2n 2 state describing the quantum computer. As far as the independent real parameters. × − coupling to the environment may be neglected, this evo- The superposition principle is clearly visible in Eq. (3): lution is unitary and governed by the Schr¨odinger equa- while n classical bits can store only a single integer k, the tion. It is well known that a small set of elementary n-qubit quantum register can be prepared in the corre- logic gates allows the implementation of any complex sponding state k of the computational basis, but also | i computation on a classical computer. This is very im- in a superposition. We stress that the number of states portant: it means that, when we change the problem, of the computational basis in this superposition can be we do not need to modify our computer hardware. For- n as large as 2 , which grows exponentially with the num- tunately, the same property remains valid for a quan- ber of qubits. The superposition principle opens up new tum computer. It turns out that, in the quantum circuit possibilities for efficient computation. When we perform model, each unitary transformation acting on a many- a computation on a classical computer, different inputs qubit system can be decomposed into (unitary) quantum require separate runs. In contrast, a quantum computer gates acting on a single qubit and a suitable (unitary) can perform a computation for exponentially many in- quantum gate acting on two qubits. Any unitary op- puts on a single run. This huge parallelism is the basis eration on a single qubit can be constructed using only of the power of quantum computation. Hadamard and phase-shift gates. The Hadamard gate is We stress that the superposition principle is not a defined as follows: it turns 0 into ( 0 + 1 )/√2 and | i | i | i uniquely quantum feature. Indeed, classical waves satis- 1 into ( 0 1 )/√2. The phase-shift gate (of phase fying the superposition principle do exist. For instance, δ| )i turns |0i −into | i 0 and 1 into eiδ 1 . Since global we may consider the wave equation for a vibrating string phases have| i no physical| i meaning,| i the states| i of the com- with fixed endpoints. It is therefore also important to putational basis, 0 and 1 , are unchanged. However, point out the role of entanglement for the power of quan- a generic single-qubit| i state| iα 0 + β 1 is mapped into tum computation, as compared to any classical compu- α 0 + βeiδ 1 . Since relative phases| i | arei observable, the tation. Entanglement is arguably the most spectacular state| i of the| qubiti has been changed by the application of and counter-intuitive manifestation of quantum mechan- the phase-shift gate. We can decompose a generic uni- ics, observed in composite quantum systems: it signifies tary transformation acting on a many-qubit state into the existence of non-local correlations between measure- a sequence of Hadamard, phase-shift and controlled-not ments performed on particles who interacted in the past (CNOT) gates, where CNOT is a two-qubit gate, defined but now are located arbitrarily far away. Mathemati- as follows: it turns 00 into 00 , 01 into 01 , 10 into cally, we say that a two-particle state ψ is entangled, or 11 and 11 into 10| .i As in| thei classical| i XOR| i gate,| i the | i non-separable, if it cannot be written as a simple product CNOT| i gate| i flips the| i state of the second (target) qubit if k1 k2 of two states describing the first and the second the first (control) qubit is in the state 1 and does noth- | i| i subsystem, respectively, but only as a superposition of ing if the first qubit is in the state 0| .i Of course, the | i such states: ψ = ck1k2 k1 k2 . For instance, the CNOT gate, in contrast to the classical XOR gate, can | i Pk1,k2 | i| i (Bell) state 1 ( 00 + 11 ) is entangled, while the state also be applied to any superposition of the computational √2 | i | i 1 ( 00 + 10 ) is separable (i.e., not entangled), since basis states.
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