An introduction to using Trapped cold Ions

March 10, 2011

Contents

1 Introduction 1

2 2

3 Operations in Quantum Computing 2 3.1 Quantum Operators ...... 2 3.2 Controlled Gates ...... 3

4 Deutsch-Jozsa Algorithm 4

5 A Trapped Ion CNOT Gate 6 5.1 Implementation of the CNOT Gate ...... 6 5.2 Rabi’s Formula and π pulses on two state systems ...... 7 5.3 CNOT Procedure ...... 8

6 Conclusion 9

1 Introduction

This paper is designed to serve as an introduction to Quantum Computing to a person having some back- ground in Quantum Mechanics and Algebra. The notion of running computers based on the mechanics of Quantum Systems was first proposed by Richard Feynman in 1982 as a proposed solution to the increasing difficulty in running simulations of quantum systems on classical computers. Through the work of Deutsch, Shore, and Grover among others [1], it has been demonstrated that a wide class of algorithms exist for which a Quantum Computer would be vastly more efficient then it’s classical analogue. Within the past decade, considerable progress has been made. In 1995, the successful implementation of a controlled-not logic gate was achieved [4] using trapped cold ions and it is this experiment that we shall discuss in great detail. More recently, implementations of far more complex quantum computations have been achieved including multiple implementations of quantum search algorithms [5] and Shor’s algorithm [6]. This paper will introduce the fundamental concepts of Quantum Computing such as qubits and quantum logic gates, in particular discussing how complex quantum (and simulations of classical) circuits may be constructed from merely a few fundamental components and briefly discussing the universlity of cnot and single qubit gates before demonstrating the Deutsch Jones algorithm, a simple algorithm demonstrating the power of quantum computation over classical systems and then describing the Experimental implementation of a CNOT gate from ultra cold atoms as achieved by Monroe et al in 1995.

1 2 Qubits

For the most part, information in computer systems has been stored in classical bits which that can take on the logical values of 0 or 1. It is upon these bits of information that classical algorithms operate. In Quantum Computing, one must introduce the notion of a qubit. Simply put, a qubit is a two state quantum system used to represent a bit of information. We may write it as

a | 0i + b | 1i (1)

For which normalization is assumed |a|2 + |b|2 = 1 (2) While a classical bit takes on one of two values, it is immediatly apparent that a qubit takes on a continuum of possible states though a single qubit, when measured, will only yield one of the two possible basis states. There are a variety of two state systems that may be used as qubits which the reader is surely familiar with. Examples include the polarization of a photon, the orientation of a spin-1/2 particle’s spin, and two energy levels of an atom are among them.

3 Operations in Quantum Computing 3.1 Quantum Operators In general, a logic gate in Quantum Computing is any unitary operator taking n qubits to n qubits (U : Cn → Cn), often referred to as “reversible” in that the inputs of a quantum gate may be determined by the outputs. In quantum circuits (computers), such gates take the place of classical logic gates that the reader may be familiar with (though for example a NOT gate does not have a quantum analogue since it is not reversible).Such operators may of course be represented as matrices. There exists several essential two qubit operators in quantum computing, several probably already familiar to anyone already with a background 0 1 0 −i 1 0  in qauntum mechanics. The Pauli matrices: σ = , σ = , and σ = are of x 1 0 z i 0 z 0 −1 course among these operators with σx filling an identical roll to the NOT gate in classical computing. Three operators that may not be so familiar are as follows.

1 1 1  H = √ (3) 2 1 −1 1 1 0 S = √ (4) 2 0 i 1 0  T = (5) 0 eiπ/4 which are known as the Hadamard, Phase, and π/8 gates respectively. The Hadamard, we will find, will prove indespensible in our discussion of the Deutsch Jones algorithm below. Explicity the effects of a Hadamard gate on the computational basis states of a single qubit system are. 1 H | 0i = √ (| 0i+ | 1i) (6) 2 1 H | 1i = √ (| 1i+ | 0i) (7) 2 Or more compactly. 1 X H | xi = √ (−1)xz | zi (8) 2 z∈{0,1}

2 These operations have their equivalences in higher order systems of multiple qubits represented quite simply as higher order 2n + 1 × 2n + 1 matrices. The Hadamard transform on n qubits is of particular importance in our discussion of the Deutsch-Josza algorithm, denoted by H⊗n.

| x . . . x i ⊗n X x1z1+x2z2+···+xnzn 1 n H | x1 . . . xni = (−1) √ (9) n n 2 (z1...zn)∈{0,1}

A final n+1 qubit operator that I will discuss is the Uf operator, or quantum oracle defined by a function f : {0, 1}n → {0, 1}.

Uf | x1, x2, . . . , xni⊗ | yi =| x1, x2, . . . , xni⊗ | y ⊕ f(x1, . . . , xn)i (10)

Where x⊕y is to be understood as addition modulo 2. This operator is unitary and is usually denoted in the circuit notation as a black box. The inner workings of an oracle are usually not drawn but we understand that as a unitary quantum operator, an oracle for any f(x) may be implemented using single qubit operators and CNOT gates due to their universality. What we mean by that sentence and what a CNOT gate is will be discussed in the following section.

3.2 Controlled Gates In computing it is often necessary to perform a tasks conditionally (’if A then do B’). In Quantum Computing, such conditional operations are encapsulated in the notion of the controlled oparator Cn(U) in which a unitary operator on k qubits acts on the condition that n qubits in it’s input are equal to one. Formally,

n x1x2...xn C (U) =| x1x2 . . . xni | ψi =| x1x2 . . . xniU | ψi (11)

x1, . . . , xn are of course assumed to take on values in {0, 1} so that U 6= Id iff x1 = 1, . . . , xn = 1. Perhaps one of the most important examples of such a controlled operator is a controlled-not (CNOT) gate, a two qubit operator with two inputs known as the control bit and the target bit. If the control bit is set to | 0i the target bit is left unchanged. Otherwise the CNOT gate flips the target bit. In the basis {| 00i, | 01i, | 10i, | 11i} the CNOT gate may be written as

1 0 0 0 0 1 0 0 CNOT =   (12) 0 0 0 1 0 0 1 0

One can imagin that as we move to higher order systems, the Matrix Notation may become a bit cumbersome to work with, especially if the behavior of each gate is well understood. It is at this point that it is useful to introduce the circuit notation often employed in the literature of Quantum Computation containing gates and wires. Quantum Circuits are intended to be read from left to right with each wire (horizontal line) corresponding to a qubit with the passage of time. Any gate is represented by a box with it’s inputs and outputs intersecting it. Controlled gates have there inputs marked by black circles. The CNOT gate is written in circuit notation in figure 1.

3 Figure 1. A CNOT gate. and as an example, the talfoli gate, which takes three qubits and flips the third on the condition that the two control qubits are equal to 1.

Figure 2. A Taffoli gate which flips the target qubit if and only if both of it’s control qubits evaluate to 1. We conclude this section by stating a result of DiVincenzo that single Qubit gates and CNOT gates are universal that is any unitary operation in quantum computation may be decomposed into sequences of operations on single qubits and CNOT operations [3]. Implementations of Quantum Circuits in such a fashion are by no means garunteed to be efficient, however this result tells us that in general, CNOT gates are all we need along with the single qubit operations to perform quantum computation, lending to the importance of the physical realization of a CNOT gate.

4 Deutsch-Jozsa Algorithm

At this point we find ourself with the notions of operators and qubits. However we are not equipped with a real notion of how these tools may be implemented to perform a task in a more efficient manner then a classical computer. The Deutsch-Jozsa algorithm, proposed in 1998, while of limited practical use serves as a good pedagogical example due to it’s relatively straightforward nature of a problem for which a quantum computational solution is exponentially more efficient at solving then it’s classical counterpart. Here we will follow a slightly cleaned up version of the algorithm that appears in Isaac and Chuang’s Quantum Computation and Quantum Information. We consider the following problem. Alice wishes to determine if a function f(x): {0, 1,... 2n −1} → {0, 1} possessed by Bob is either constant or balanced (the function is guaranteed to be one or the other). By

4 Constant we mean of course that f(x) = 1 XOR f(x) = 0 ∀x ∈ {0,..., 2n − 1} and by balanced we mean that f(x) = 0 for exactly half of the x in the domain. The means by which Alice is allowed to determine the nature of Bob’s function is to send him values of x to which he is obliged to provide her with the appropriate f(x). At worst, given this system in the classical case, Alice would be unlucky enough to sample from the exact half of the x’s in the domain that give the same f(x), meaning 2n/2 + 1 trials before, if f is balanced she will recieve an answer different from the previous ones and if f is constant she will receive an answer identical to her previous queries. A quantum solution to the problem, the Deutsch-Jozsa algorithm, exists which involves only one iteration, whose circuit is displayed in figure 3.

Figure 3. The for the Deutsch-Jozsa algorithm. Alice performs n qubits by performing a parallel Hadamard Transform on | 0i⊗n and a seperate Hadamard transform on the target qubit, | 1i. Bob applies the Oracle and Alice performs a Hadamard on the output before performing her measurement. In essence, any of the x inputs to f(x) are encoded in n qubits, so Alice’s input is a system of n qubits. Alice also posseses a single qubit which she will provide to Bob to place the output of f(x) in. Further, Bob has agreed to evaluate Alice’s query using a quantum oracle Uf as described in equation 10. Rather then send queries to Bob blindly, Alice will start out with | 0i⊗n as her inputs for f upon which she will perform an H⊗n operation on. Further, she will prepare the qubit that Bob must use as the target bit for the oracle |0i−|1i by preparing a hadamard transformed H | 1i = √ . In sum, Alice starts things off by providing Bob the 2 following system. We evaluate the effects of the Hadamard transformation based on equation 9.

⊗n ⊗n | ψprepi = H | 0i ⊗ (H | 01i) (13)   X | z1 . . . zni | 0i− | 1i = √ ⊗ √ (14) n n 2 2 (z1,...,zn)∈{0,1} X | xi | 0i− | 1i = √ ⊗ √ (15) 2n 2 x∈{0,1}n

Now lets look at how the Oracle behaves when it is applied to Alice’s prepared bits.

( |0i−|1i √ f(x) = 0 | 0 + f(x)i− | 1 + f(x)i 2 √ = |0i−|1i (16) 2 − √ f(x) = 1 2 | 0i− | 1i = (−1)f(x) √ (17) 2 Thus the output that Bob provides with the Oracle is

X (−1)f(x) | xi | 0i− | 1i | ψbobouti = √ ⊗ √ (18) 2n 2 x∈{0,1}n

5 As the last step, Alice takes her n input bits and applies yet another Hadamard transform on them. After this, the final state, which Alice will perform measurements on is

f(x) !   ⊗n X (−1) | xi | 0i− | 1i | ψfinali = H √ ⊗ √ (19) n x 2 2 X X (−1)x1z1+···+xnzn + f(x) | 0i− | 1i = √ (20) 2n 2 z∈{0,1}n x∈{0,1}n

⊗n ⊗n Now consider C0 = h0 | ψfinalin, the | 0i component of the n output bits that Alice is left with. If f is balanced, then clearly X (−1)f(x) C = = 0 (21) 0 2n x and if f is constant ( X (−1)f(x) 1 f(x) = 0 C = = (22) 0 2n x −1 f(x) = 1 So all Alice has to do in the end is measure the n input bits that she originally fed into the system. If she measures all of her qubits to be in the | 0i state then the function was constant. On the other hand, if she measures none of her qubits to be in the | 0i state, the function was balanced. All in all, the number of iterations the algorithm had to undergo was 1, so that it was completed in constant time whereas we saw above that a classical approach would execute in on the order of 2n operations. The power that this technique derives itself from is that while we may only observe one of the computational basis states, an unobserved system may carry out computations on all possible values for qubits simultaneously, the setup of the problem (that f was either constant or balanced) was such that only one observation was necessary on each of the input qubits to determine the solution with certainty. In general three families of quantum algorithms to date have been discovered that offer considerably greater efficiency then any known classical solutions: Quantum Searches, Quantum Fourier Transforms, a family in which the famous Shor’s algorithm is a part of, and Quantum Simulations. Unfortunately these algorithms are beyond the scope of this paper thought their importance makes them worth mentioning here. Quantum Computers can be used to solve a certain class of problems far more efficiently then classical computers, but how might we actually implement them? We mentioned briefly what physical objects might be used as qubits however it is unclear how we might perform the unitary operations on these qubits that were described in the previous sections. A physically realized quantum circuit, the CNOT gate, will now be discussed.

5 A Trapped Ion CNOT Gate 5.1 Implementation of the CNOT Gate We now describe the realization of the CNOT gate by Monroe et al. The Hydrogen-like ion,9Be+ is confined by a rapidly oscillating RF potential in a coaxial resonator rf ion trap [4] so that they experience a motional hamiltonian N  2  N 2 X M 2 2 2 2 2 2 |pi| X X e H = ωxxi + ωyyi + ωz zi + 2 + (23) 2 M 4π0|~ri − ~rj| i=1 i=1 j6=i

hence given that the atoms are sufficiently isolated from their environment and cooled s.t. kBT << ~ωx the x y and z motional degrees of freedom of the ions are described as quantized harmonic oscillators. The design of the trap garuntees that ωz, ωy >> ωx. For the CNOT gate, the target qubit is spanned by the two 2 S1/2 hyperfine ground states of the Beryllium ion: | F = 1, mF = 1i and | F = 2, mF = 2i and the control qubit is spanned by the first two motional ground states in thex ˆ direction | nx = 0i and | nx = 1i. The qubit correspondence is given the below table.

6 Physical State Shorthand | nx = 0i | F = 2, mF = 2i | 0i |↑i | nx = 1i | F = 2, mF = 2i | 1i |↑i | nx = 0i | F = 1, mF = 1i | 0i |↓i | nx = 1i | F = 1, mF = 1i | 1i |↓i Table 1. The Physical states used in the Trapped ion quantum computer and their corresponding computational basis representations.

Figure 3. The Energy Levels of the 9Be+ used in the implementation of the CNOT gate. Image borrowed directly from [4]

Energy levels used for this system are illustrated in figure 3. The |↑i and |↓i states of the Be ion are seperated by ω0/2π = 1.250GHz and the two motional states, | 0i and | 1i are separated by ωx/2π = 11MHz. Three types of transitions are stimulated between the states via coherent excitations by the frequency difference of two off resonant lasers whose frequency differences are denoted by δ. The carrier transmission is meant to effect only the target bit while the red and blue side band transitions mix between the target and carrier control bits. In particular, we will implement a controlled Z on the blue sideband. Transition Energy Name | 0i |↑i ←→| 0i |↓i δ = ω0 carrier | 1i |↑i ←→| 1i |↓i δ = ω0 carrier | 1i |↓i ←→| 0i |↑i δ = ω0 − ωx red sideband | 0i |↓i ←→| 1i |↑i δ = ω0 + ωx blue sideband Table 2. The transitions stimulated by application to the single ion system by off resonant laser pulses. Before we discuss by what means in this setup, the CNOT gate is actually executed, it is worth digressing to discuss π pulses and Rabi’s formula.

5.2 Rabi’s Formula and π pulses on two state systems To understand on a basic level how the experimenters are able to physically perform quantum operations on their qubits, recall the two state system from time dependent perturbation theory spanned by {| 1i, | 2i iωt . For a potential with harmonic time dependence, V = V0e as is the case for linearly polarized coherent light perturbing a hydrogen like ion. We may consider a toy example that appears in Sakurai [7].

V = γeiωt | 1i h2i + γ∗e−iωt | 2ih1 | (24)

7 which is a reasonable hamiltonian if we assume the dipole approximation with ω real. If we assume that the perturbation is on resonance, ω12 = ω as is the case in all of our two state transitions, we may use time dependent perturbation theory to arrive at the very simple equations

i~c˙1 = γc2 (25) ∗ i~c˙2 = γ c1 (26) We may substitute both equations into eachother. For some familiar second order differential equations

γ2 c¨1 = − c1 (27) ~2 γ2 c¨2 = − c2 (28) ~2 With familiar solutions (from now on let γ denote the magnitude of γ, a real number). γ γ c1(t) = A1Cos t + B1Sin t (29) ~ ~ γ γ c2(t) = A2Cos t + B2Sin t (30) ~ ~

Equations 27 and 28 constrain the coefficients to be related by B2 = −iA1 and A2 = iB1. The solutions may finally be written in the form Ω Ω c (t) = ACos t + BSin t (31) 1 2 2 Ω Ω c (t) = iBCos t − iASin t (32) 2 2 2 We refer to Ω in this case as the Rabi frequency for the system. Note that for a system starting with c1(0) = 1 and c2(0) = 0 we have A = 1, B = 1. If we apply the perturbation for the time t = 2π/Ω we will transform | 1i → − | 1i. A laser pulsed for this time length is known as a 2π pulse and can be used to shift the phase of a save function by π. On the other hand, given the same initial conditions a pulse |1i−|2i for t = π/(2Ω) will place the system in a superposition √ . This sort of pulse is known as a π/2 pulse. 2 Finally, application of the perturbation for t = π/Ω will transform | 1i → −i | 2i. Unfortunately we cannot simply apply this toy formalism to our system (the author has tried and ended up performing CNOT operations with factors of i and -i introduced in the end, our quantum computer is sensitive to phases, constructing and destructive interference for example are the cornerstones of the power of the Deutsch-Jozsa algorithm discussed earlier). The hamiltonian in which the atoms reside is a perturbed Jaynes-Cummings Hamiltonian which requires a fully quantized electric field to understand. The reader is referred to [2] for a thorough theoretical characterization of the of the laser interaction.

5.3 CNOT Procedure The procedure for CNOT implementation in [4] goes as follows. A Raman pulse is applied to the ion to prepare an arbitrary state in the register.

1. Next, a π/2 pulse is applied to the carrier transition. This is the equivalent of applying Ry(π/2) [1] to the target qubit.

2 2. , A 2π pulse is applied between the | 0i | auxi state and | 1i |↑i. Where | auxi is the S1/2 | F = 2, mF = 0i ground state split from |↓i by Zeeman shift due to an applied 0.18 mT magnetic field. The effect of this pulse is to take | 1i |↑i →| 0i | auxi → − | 1i |↑i effectively this is a controlled Z gate with the control bit in this case being the target qubit.

8 3. , Finally, another π/2 laser pulse is applied to the carrier transition with a π phase shift making it correspond to R(−π/2) [1].

Figure 3. The sequence of operations that constitute the trapped ion CNOT gate implemented by Monroe et al.

The effective circuit formed by these operations is displayed in figure 4. Let’s compute the truth table 1 −1  1 1 now by considering R(π/2) = √1 and R(−π/2) = √1 . 2 1 1 2 −1 1

Id⊗R (π/2) |↑i+ |↓i Id⊗R (−π/2) |↑i− |↓i+ |↑i+ |↓i | 0i |↑i −−−−−−−−→|y 0i √ −−−−−−−−−→|y 0i =| 0i |↑i (33) 2 2 Id⊗R (π/2) |↓i− |↑i Id⊗R (−π/2) |↓i+ |↑i− |↑i+ |↓i | 0i |↓i −−−−−−−−→|y 0i √ −−−−−−−−−→|y 0i =| 0i |↓i (34) 2 2 Id⊗R (π/2) |↑i+ |↓i |↓i− |↑i Id⊗R (−π/2) |↑i+ |↓i− |↑i+ |↓i | 1i |↑i −−−−−−−−→|y 1i √ to | 1i √ −−−−−−−−−→|y 1i =| 1i |↓i (35) 2 2 2 Id⊗R (π/2) |↓i− |↑i |↓i+ |↑i Id⊗R (−π/2) |↑i+ |↓i+ |↑i− |↓i | 1i |↓i −−−−−−−−→|y 1i √ →| 1i √ −−−−−−−−−→|y 1i =| 1i |↑i (36) 2 2 2 Clearly the result of the three pulses on a given qubit is a controlled not gate. The authors, upon performing their operation, verified the output qubit by taking a measurement, first of |↑i of the target bit. 2 A laser resonant with the P3/2 | F = 3, mF = 3i ←→|↓i transition is applied with fluoresence indicating the P (|↓i). If the state is known to be in the |↓i state after this measurement, a π pulse is applied to the red sideband. If no fluorescence was measured, the authors determined the ion to be in the nx = 1 state as can plainly be seen from Table 2. Otherwise nx = 0 must be occupied (so that the red side band does not excite a transition). On the other hand, if the atom was known to be in the |↑i state a π pulse is applied to the blue sideband. From Table 2 it is apparent that a lack of fluoresence implies the nx = 0 state. As a final note, the coherence time, essentially how long a state can remain in the system without being destroyed by outside noise and interference, putting a cieling on the length of quantum operations one might hope to perform on them was also measured in the paper. The author’s determine it to be 50µsec

6 Conclusion

As can be seen, Quantum computation via Unitary manipulations on qubits offers in certain situations, a vastly more efficient way of solving problems. Universality ensures that single bit qubit operations and CNOT gates can be combined to perform arbitrarily complex operations on any number of qubits. Recent experiment, as discussed in this paper, has realized the implementation of a single qubit CNOT gate. More recent experiments have suceeded in creating quantum computations of great complexity including Shor’s algorithm on several qubits. The Author hopes that you have found this paper clear and informative.

9 References

[1] Chuang and Nielson, Quantum computation and quantum information, Cambridge University Press, New York, NY, 2000. [2] Cirac and Zoller, Quantum computations with cold trapped ions, Physical Review Letter 74 (1995), no. 20. [3] DiVincezo, Two-bit gates are universal for quantum computation, Physical Review Letters A 51 (1995), no. 2.

[4] Monroe et al, Demonstration of a fundamental , Physical Review Letters 75 (1995), no. 25. [5] Vandersypen et al, Implementation of a three-quantum-bit search algorithm, Applied Physics Letters 76 (2000), no. 5.

[6] , Experimental realization of shor’s quantumf factoring algorithm usng nuclear magnetic reso- nance, Nature 414 (2001). [7] Sakurai and Neapolitino, Modern quantum mechanics, Pearson, New York, NY, 2010.

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