Fidelity of geometric and holonomic quantum gates for spin systems

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UPTEC F 14012

Examensarbete 30 hp April 2014

Fidelity of geometric and holonomic quantum gates for spin systems

Daniel Töyrä

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Teknisk- naturvetenskaplig fakultet UTH-enheten

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Abstract

Fidelity of geometric and holonomic quantum gates for spin systems

Daniel Töyrä

Geometric and holonomic quantum gates perform transformations that only depend on the geometry of a loop covered by the parameters controlling the gate. These gates require adiabatic time evolution, which is achieved in the limit when the loop takes infinite time to complete. However, it is of interest to also know the

transformation properties of the gates for finite run times. It has been shown [Phys.

Rev. A 73, 022327 (2006)] that some holonomic gates for a trapped ion system show revival structures, i.e., for some finite run time the gate performs the same

transformation as it does in the adiabatic limit.

The purpose of this thesis is to investigate if similar revival structures are shown also for geometric and holonomic quantum gates for spin systems. To study geometric quantum gates an NMR setup for spin-1/2 particles is used, while an NQR setup for spin-3/2 particles is used to study holonomic quantum gates. Furthermore, for the geometric quantum gates the impact of some open system effects are examined by using the quantum jump approach. The non-adiabatic time evolution operators of the systems are calculated and compared to the corresponding adiabatic time evolution operators by computing their operator fidelity. The operator fidelity ranges between 0 and 1, where 1 means that the gates are identical up to an unimportant phase factor. All gates show an oscillating dependency on the run time, and some Abelian gates even show true revivals, i.e., the operator fidelity reaches 1.

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Populärvetenskaplig sammanfattning

Kvantdatorer har varit på tapeten sedan Peter Shor presenterade en kvantalgoritm som primtalsfaktoriserar ett givet heltal mycket eektivare än någon känd klassisk algoritm.

Många av dagens krypteringssystem är byggda på att det är svårt för en klassisk dator att primtalsfaktorisera stora heltal, och dessa krypteringssystem skulle bryta samman om man lyckades bygga en tillräckligt kraftfull kvantdator. I gengäld kan dock kvantdatorer erbjuda nya och säkrare krypteringssystem. Kvantdatorer har även potential att över- glänsa klassiska datorer på andra uppgifter, såsom att söka i databaser och att simulera kvantmekaniska processer.

Det största hindret för att kunna bygga kraftfulla kvantdatorer är att systemet måste isoleras från omgivningen tillräckligt väl och tillräckligt länge för att beräkningar ska hinna utföras. Växelverkan med omgivningen skapar brus i systemet och för mycket brus innebär att informationen i systemet går förlorad.

I en klassisk dator kallas informationsbärarna för bitar, där en bit antingen har värdet 0 eller 1. I en kvantdator kallas informationsbärarna för kvantbitar, där en kvantbit i allmänhet är i en superposition av de kvantmekaniska tillstånd som representerar 0 och 1. Med andra ord kan en kvantbit vara både 0 och 1 samtidigt. Om man har två bitar och två kvantbitar så är bitarna i ett av de fyra möjliga tillstånden, medan kvantbitarna kan vara i alla fyra tillstånden samtidigt. Har man istället tre bitar och tre kvantbitar så är bitarna i ett av de nio möjliga tillstånden medan kvantbitarna kan vara i alla nio tillstånden på en gång. Och så här kan man hålla på så länge man vill, uppenbarligen blir det större och större skillnad mellan en kvantdator och en klassisk dator ju större systemet blir. I det här examensarbetet betraktas kvantbitar som består av spinnpartiklar, där 0 och 1 är inkodade som upp- respektive nedspinn längs någon vald riktning.

Kvantgrindar är byggstenarna i en kvantdator. De utför operationer på kvantbitar lik- som klassiska logiska grindar bestående av transistorer och/eller dioder gör på bitar i en klassisk dator. Kvantgrindar som opererar på spinnpartiklar använder sig lämpligen av magnetfält.

1984 visade den engelske fysikern Sir Michael Berry att kvantmekaniska tillstånd som långsamt transporteras runt en sluten kurva erhåller ett minne av kurvan i form av en geometrisk fasfaktor. Detta innebär att tillståndet efter transporten har förändrats på ett sätt som beror på den slutna kurvans geometri. Upptäckten var häpnadsväckande, generationer av fysiker hade missat det Berry elegant satte ngret på. För spinnpartiklar i ett magnetfält beror den geometriska fasfaktorn på rymdvinkeln som inkapslas av magnetfältets slutna kurva.

Dynamiska fasfaktorer beror på magnetfältets styrka och riktning, samt exponeringens längd. Vanligtvis används dynamiska fasfaktorer för att bygga kvantgrindar. Nackdelen

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med detta är att dynamiska faser är känsliga för lokala avvikelser hos magnetfältet. Den geometriska fasen däremot, beror på globala egenskaper (rymdvinkeln) hos den slutna kurvan. Om uktuationerna är små och någorlunda slumpmässiga så kommer kurvans rymdvinkel i genomsnitt vara nästintill oförändrad eftersom de slumpmässiga variation- erna tar ut varandra längs kurvan. Detta är grundtanken med att bygga kvantgrindar baserade på geometriska faser.

I det här examensarbetet har två typer av grindar undersökts, den ena typen kallas geometrisk kvantgrind och bygger på Berrys ursprungliga geometriska fas. Den andra typen kallas holonom kvantgrind och använder en generellare och lite mer komplicerad geometrisk fasfaktor, men som bygger på samma geometriska princip. Båda dessa typer av grindar kräver dock adiabatisk tidsutveckling, vilket innebär att färden längs den slutna kurvan tar oändligt lång tid. Om grindarna ska implementeras är det dock bra att även känna till deras egenskaper för ändlig körtid. Därför har grindarnas transfor- mationsegenskaper även beräknats för varierande ändliga körtider, och sedan har dessa jämförts med de ideala adiabatiska transformationerna genom att beräkna deliteten.

Fideliteten är ett reellt tal mellan 0 och 1, och mäter transformationernas likhet där 1 motsvarar identiska transformationer (upp till en oviktig global fasfaktor).

Som väntat går deliteten mot 1 när körtiden går mot oändligheten (adiabatiska gränsen), men desto mer intressant är att deliteten oscillerar, och för vissa grindar når delitet- stopparna 1 även för ändliga körtider. Detta innebär att om man skulle kunna ställa in körtiden till en av dessa toppar så reduceras körtiden och därmed också risken för att omgivningen ska hinna skapa tillräckligt mycket brus i systemet för att förstöra informationen.

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List of Figures

1. Expectation value of a spin-¹₂ vector. . . 11

2. Parallel transport of a vector on the surface of a sphere. . . 14

3. A qubit represented on the Bloch sphere. . . 15

4. The spin echo scheme used for geometric quantum gates . . . 21

5. The general closed path used for holonomic quantum gates. . . 34

6. Operator delity of four geometric quantum gates with phase damping. . . 41 7. Operator delity of four geometric quantum gates with amplitude damping. 42

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8. Operator delity of four Abelian holonomic quantum gates . . . 44 9. Operator delity of four non-Abelian holonomic quantum gates . . . 45

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1. Introduction

In 1984 Berry [1] showed that adiabatic time evolution¹ of a non-degenerate eigenstate of a periodic Hamiltonian contains a purely geometric phase factor in addition to the well known dynamical phase factor. This means that the evolved eigenstate contains a complex number of unit norm, whose phase carries geometric information about the traversed path. This phase factor is Abelian (commutative) because multiplication is a commutative operation for complex numbers. Even though it was long known that this extra phase factor arises in the calculations, Berry was the rst to realise its physical signicance. Generations of physicists thought that this phase factor could be absorbed by the eigenvectors [2], but Berry demonstrated that there is generally not possible to do so for cyclically evolving eigenstates.

However, geometric phase factors had been encountered in physics long before. In the late 30s and the early 40s Rytov [3] and Vladimirskii [4] studied how the electric and magnetic elds of light rays are parallel transported in inhomogeneous media. Even though they did not explicitly point it out, their work inferred phase holonomy² for circularly polarised light-rays [5]. In 1956, Pancharatnam [6] showed that a cyclic change in the polarization state of light gives rise to a phase shift of geometric origin. Berry [7], and Ramaseshan and Nityananda [8] worked out the relation between the quantum mechanical Berry phase and the optical Pancharatnam phase and showed that Pancharatnam's phase is an optical analogue of the Aharonov-Bohm eect [9]. In 1958 Higgins et al. [10]

discovered a sign change of the wavefunction when the internal coordinate system of a molecule undergoes a loop, and Herzberg and Longuet-Higgins [11] interpreted this as a holonomy in 1963.

Simon [12] connected Berry's discovery to the mathematical eld of dierential geometry, and interpreted Berry's phase as a holonomy of a complex line bundle. Berry's Abelian phase factor was generalised to degenerate eigenstates by Wilczek and Zee [13], whose phase factor is in fact matrix-valued, i.e., potentially non-Abelian. The concept of geometric phase has been further generalised. For example, Aharonov and Anandan [14]

showed that the Abelian phase factor does not need to be adiabatic, Anandan [15] did the same thing for the non-Abelian phase factor, and Samuel and Bhandari [16] even removed the need of cyclic evolution for the Abelian phase by using an idea borrowed from Pancharatnam.

The concept of quantum computation was introduced in the 70's, and theoretical quantum computers were proposed by pioneers such as Feynman, Benio and Deutsch in the 80's. But following Shor's [17] demonstration of his prime factoring algorithm in 1994, the interest in quantum computation increased heavily. His algorithm would be able to prime factorise large integers more eciently than any known algorithm for classical com-

1 Slowly varying Hamiltonian, see section 2.3.

2 Holonomy is a geometric property, see section 2.4.

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puters, and consequently, be capable of breaking the widely used RSA cryptosystem [18, p. 11]. In return, quantum cryptography [18, pp. 582-602] oers new and safer methods for encrypting information. Quantum computers are also assumed to outperform classical computers in some other tasks, such as simulating physical quantum systems [18, pp.

39-40] and searching in a database [18, 248-276].

In 1999 Zanardi and Rasetti [19] theoretically showed that Wilczek and Zee's non-Abelian geometric phase factor can be used for quantum computation, and thereby founded holonomic quantum computation [20]. The idea is to encode a computational system in a degenerate eigenspace of a parameter dependent Hamiltonian, and to let these parameters slowly travel around a loop in parameter space. When the parameters have completed the loop, the initial state has evolved into a dierent nal state in the same eigenspace.

The transformation from the initial to the nal state only depends on the loop's geometry in parameter space. This transformation constitutes the holonomic quantum gate.

Geometric quantum computation is based on Berry's Abelian geometric phase factor. It uses the same principle as holonomic quantum computation, but with some signicant dierences. The initial state is prepared in a superposition of two non-degenerate eigenstates, and it is the relative geometric phase factor of the two, independently evolved, eigenstates that denes the transformation. In 1999, Jones et al. [21] experimentally demonstrated geometric quantum computation by using a nuclear magnetic resonance (NMR) setup.

Quantum gates built on geometric phases are generally assumed to be resilient against errors from randomly uctuating parameters [20]. The reason for this is that geometric phases only depend on global properties of the adiabatic evolutions, while the uctua- tions are local. For example, the Abelian geometric phase obtained by a cyclically evolved spin-¹₂ eigenstate is given by half the solid angle enclosed by the loop in parameter space.

The random errors are evened out around the loop so the solid angle will on average be unaected by the errors.

Since Berry and Wilczek-Zee phase factors appear under adiabatic time evolution, the run time of gates based on these phase factors must be long enough for the adiabatic approximation to be valid. This is a problem because the probability for decoherence increases with increasing run time. To reduce the run time, gates built on non-adiabatic geometric phase factors have been proposed. Wang and Matsumoto [22] proposed non- adiabatic conditional geometric phase shift gates, using the Aharonov-Anandan phase factor [14], and Sjöqvist et al. [23] proposed universal non-adiabatic holonomic quantum computation that makes use of Anandan's non-Abelian phase factor [15].

This thesis focuses on geometric and holonomic quantum gates, both based on adiabatic time evolution. However, adiabatic time evolution is an approximation, real systems never evolve completely adiabatically. Therefore, it is necessary to investigate the exact

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behaviour of the gates, i.e., without the adiabatic approximation. The subsequent anal- ysis is made by looking at the resemblance between the ideal adiabatic target gate and the exact gate while varying the run times. The resemblance is measured in operator

delity, which falls in the range [0, 1], where 1 means that the gates are identical up to an unimportant overall phase factor. Naturally, the operator delity should approach 1 as the run time goes to innity (the adiabatic limit). However, it is known from Florio et al. [24] that tripod systems [25] show revival structure, which means nite run times that yield unit delity. Thus, by operating a gate at its revival with shortest run time, the risk of decoherence is minimized.

The purpose of this thesis is to investigate if delity revivals are shown also for spin systems. In section 3, geometric quantum gates for qubits³ consisting of spin-¹₂ particles are studied. The time evolution operator that describes the gate-transformation is calculated analytically, both exactly and adiabatically, and the impact of some open system eects are analysed. In section 4, a holonomic quantum gate operating on qubits encoded in degenerate states of spin-³₂ particles are studied. The time evolution operator that char- acterises the gate is analytically calculated both under the adiabatic approximation and in exact non-adiabatic treatment. The resulting operator delities for the two dierent types of gates are presented in section 5. But rst, some concepts used in this thesis are described in section 2.

3 Quantum bits, see 2.5

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2. Notations and important concepts

In this section, notations and important concepts that are frequently used in this thesis are described.

2.1. Notations

A general pure spin state is denoted |αi, and if such a state is evolved in time from t0 to t, it is written |α, t0; ti =U (t, t0) |α, t₀; t₀i, where U (t, t0)is the time evolution operator.

When spin-¹₂ systems are considered, the eigenstates of the spin operator ˆn · S are denoted |ˆn · S; ±i, where S = ˆxSx + ˆyS_y + ˆzS_z is the spin vector, ˆn is a unit vector, and the sign indicates if the spin is parallel (+) or anti-parallel (−) to ˆn. Thus, the eigenvalue equation ˆn · S |ˆn · S; ±i = ±^~₂|ˆn · S; ±i is satised. For spin-³₂ systems, the corresponding eigenstates are denoted |ˆn · S; mi, where m ∈ −³₂, . . . ,³₂

. Hence, the eigenvalue equation ˆn · S |ˆn · S; mi = m~ |ˆn · S; mi is satised. For both spin systems, the eigenstates of the operator Sz are given special notations: |±i for spin-¹₂ and |mi for spin-³₂. Instantaneous eigenstates of a time dependent Hamiltonian H(t) are denoted by the Greek letter η, i.e., these states satisfy H(t) |η_k(t)i = Ek(t) |η_k(t)i, where Ek(t)are energy eigenvalues.

Rotations are performed by the rotation operator D(ˆn, φ) = exp

−iφˆn·S

~ , where φ is the rotation angle and ˆn is the axis of rotation. Rotations about the x, y and z axes are given special notations, for example, Dz(φ) =D(ˆz, φ).

The equality sign with a dot (.

=) is frequently used in this thesis, and it should be read as

"...is represented by the matrix...". Furthermore, the operators that are used and their matrix representations are listed in appendix A.

2.2. Overall and relative phase factors

A physical state in quantum mechanics is only dened up to a phase factor, that is, a complex number of norm 1. Thus, the two states |αi and e^iφ|αi, where φ ∈ R, correspond to the same physical state even though they have dierent representations in complex Hilbert space. This type of phase factor is called overall phase factor and it does not have any physical signicance. Now, consider a spin-¹₂ particle that has no other degrees of freedom besides its spin. It can be shown that any spin-¹₂ state can be obtained by rotating |+i twice: rst around the y-axis with an angle θ ∈ [0, π], and subsequently around the z-axis with an angle ϕ ∈ [0, 2π]. By using the rotation operators, any normalised state can be parametrised as

|αi =Dz(ϕ)Dy(θ) |+i

= cos (θ/2) e^−iϕ/2|+i + sin (θ/2) e^iϕ/2|−i

= e^−iϕ/2 cos (θ/2) |+i + sin (θ/2) e^iϕ|−i , (1)

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where the overall phase factor e^−iϕ/2in the last line can be dropped. This parametrisation is not only valid for spin-¹₂ systems, but for all two-level systems. Clearly, a relative phase factor e^iϕ between the two basis states has emerged. To demonstrate the signicance of this relative phase let us compute the expectation value of the spin vector,

hSi = hα|S|αi = ~

2(ˆx sin θ cos ϕ + ˆy sin θ sin ϕ + ˆz cos θ) . (2) Obviously, the relative phase ϕ determines the relative probability to measure the spin in the x and y directions, as visualised in gure 1. Thus, even though overall phase factors are non-physical, relative phase factors are denitely of physical signicance.

2.3. Adiabatic time evolution and geometric phase factors

Consider a Hamiltonian H R(t) that depends continuously on the parameters R(t) = (r₁(t), r₂(t), . . . ), and that at any instant has K distinct energy eigenvalues Ek R(t) where k = 1, 2, . . . , K. For each Ek R(t) ,

there is an orthonormal set Ak R(t) consisting of Nkdegenerate instantaneous eigenstates |ηa^k R(t)iof the Hamiltonian, where a = 1, 2, . . . , N_k. Thus, Ak R(t)

is dened by

A_k R(t) = |η^k_a R(t)i : H R(t) |η^k_a R(t)i = Ek R(t) |η^k_a R(t)i ,

hη_a^k R(t)|η^k_b R(t)i = δab . (3) Hence, each set Ak R(t)

forms a degenerate eigenspace with degree of degeneracy Nk. At any instant, the union S^K_k=1A_k R(t)

constitutes a complete basis, thus a general state |αi can at any instant be expressed in terms of the instantaneous eigenstates. Thus, a state that is evolved in time from t0 to t can be written

|α, t0; t0i =

K

X

k=1 N_k

X

a=1

c^k_a(t0) |η_a^k R(t0)i → |α, t0; ti =

K

X

k=1 N_k

X

a=1

c^k_a(t) |η_a^k R(t)i , (4)

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where c^k_a(t) = hη_a^k R(t)|α, t0; ti. In the adiabatic approximation, the eigenspaces are approximately isolated from each other during the time evolution [26], that is, the probability of nding the state in eigenspace k is preserved throughout the evolution. This means that the probability coecients satisfy

Nk

X

a=1

c

ka(t)

2 ≈

Nk

X

a=1

c

ka(t₀)

2. (5)

The adiabatic theorem states that if the Hamiltonian changes slowly compared to a char- acteristic internal time scale, then the system evolves according to the adiabatic approximation [26]. Assuming that the adiabatic approximation is valid, the Schrödinger equation can be solved separately for each eigenspace. By using equation (3) the Schrödinger equation for the kth eigenspace is

i~∂

∂t

Nk

X

a=1

c^k_a(t) |η^k_a R(t)i = E_k R(t)

Nk

X

a=1

c^k_a(t) |η_a^k R(t)i . (6) From now on the index k is omitted since the same arbitrary eigenspace is considered throughout this discussion. By taking the inner product of both sides of equation (6) with |ηb R(t)i, one obtains

dc_b(t) dt = −i

~E R(t)cb(t) −

N

X

a=1

c_a(t) hη_b R(t)| d

dt|η_a R(t)i . (7) By introducing the vector potential Aab(R) = i hηa(R)| ∇R|η_b(R)i, equation (7) can be written

dc_b(t) dt = −i

~E R(t)cb(t) + i

N

X

a=1

c_a(t)A_ba R(t) ·dR(t)

dt , (8)

which in matrix form has the solution

c(t) = exp

−i

~ Z t

t0

E(t⁰)dt⁰

P exp i Z R(t)

R(t0)

A_jdR^j

!

c(t₀). (9) Here, P is the path-ordering operator,

c(t) =





 c₁(t) c2(t)

...

cN(t)







, (10)

and

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A_j =







A11(R)

j A12(R)

j · · · A1N(R)

j

A21(R)

j A22(R)

j · · · A2N(R)

... ... ... ... j

A_{N 1}(R)

j A_{N 2}(R)

j · · · A_{N N}(R)

j







. (11)

The rst exponential in equation (9) is the usual dynamical phase factor while the second is of purely geometric origin. Note that, generally, c(t) and c(t0) are sets of expansion coecients for dierent bases, i.e., |ηa R(t)i 6= |ηa R(t0)i . But if the path makes a loop C in parameter space and the instantaneous eigenbasis is single-valued around C, then the initial and nal bases are equal. In this case, the geometric part of equation (9) becomes Wilczek and Zee's [13] geometric phase factor

U_{W Z}(C) =P exp i

Z _R(t₀_{+T )}

R(t0)

A_jdR^j

!

=P exp

i

I

C

A_jdR^j

, (12)

where T is the time it takes to traverse C, consequently R(t0+ T ) = R(t0). For a given system, the geometric phase factor UW Z(C)only depends on the loop and not on how the loop is traversed (as long as the adiabatic approximation is valid). For a non-degenerate instantaneous eigenstate |η R(t)i, equation (12) reduces to Berry's [1] original phase factor

U_B(C) = exp

i

I

C

A(R) · dR

, (13)

where A(R) = i hη(R)| ∇R|η(R)i.

This section is concluded with a remark: it is possible to choose a basis that is changed after a cyclic evolution, and for non-cyclic evolutions it is not even trivial to dene what one means with equal initial and nal bases. How these cases are treated can be found in [27].

2.4. Holonomy

Holonomy is an eect that arises in dierential geometry when geometrical data is parallel transported around a closed loop on a curved surface. An example is the parallel transport of a vector around a closed loop on the surface of a sphere. Consider a vector, initially located in the xz-plane on the equator and pointing in the positive z-direction.

The vector will be transported to the north pole along a geodesic, back to the equator along another geodesic and nally back to its original position along a third geodesic (the equator). The journey of the vector is visualized in gure 2. The rule for parallel transport is that the vector is forbidden to be locally changed. When the vector returns

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after its journey it nds itself rotated compared to its initial direction, even though it never underwent any local rotation. The rotation is instead a consequence of the curved surface of the sphere. Holonomy measures the failure to preserve parallel transported geometrical data around a closed loop, in this example the holonomy is characterized by the angle ϕ in gure 2. In quantum mechanics, geometric phase factors, both in scalar and matrix forms, are consequences of Holonomies and they reveal the non-trivial curvature of the state space.

Figure 2: A vector is parallel transported around a loop on the surface of a sphere. Even though the vector never is locally rotated, it ends up rotated relative to its initial direction.

2.5. Qubits, quantum gates, and universality

Qubits are the information carriers of quantum computers, just as bits are the information carriers of classical computers. While a classical bit is either 0 or 1, a qubit can be in any superposition of the quantum states representing 0 and 1, that is, a qubit can represent both 0 and 1 at the same time. If we have k bits and k qubits, the bits are in one of the 2^k possible permutations while the qubits can be in all of the 2^k possible permutations at the same time.

An arbitrary qubit can be written as |αi = A |0i + B |1i, where A and B are complex numbers satisfying |A|²+ |B|² = 1. In matrix form this qubit is written |αi .

= (A B)^T, and according to equation (1) the coecients can always be parametrised as (A B)^T =

cos(θ/2) sin(θ/2)e^iϕT

, up to an unimportant overall phase factor. Qubits are often geometrically represented on the Bloch sphere, as seen in gure 3. The north pole corresponds to 0 and the south pole corresponds to 1, i.e., a classical bit is restricted to the poles. But for a qubit, each point on the surface of the Bloch sphere represents a possible state. In this thesis the qubits consist of spin particles where 0 and 1 are encoded as spin up and spin down relative to some direction. In gure 3, this direction is the z-axis.

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Figure 3: A qubit given by |αi = cos (θ/2) |0i + sin (θ/2) e^iϕ|1i is represented on the Bloch sphere. The surface of radius 1 represents pure states while the interior represents mixed states.

In a quantum computer an input register of qubits is evolved unitarily by a set of quantum gates to yield an output register. The output is either read out or used as an input to a new set of gates. One such quantum gate is the Hadamard gate, which is dened as

H .

= 1

√2

1 1

1 −1

, (14)

where the matrix is expressed in the basis {|0i , |1i}, in that order. This gate is a one- qubit gate, that is, it operates on one qubit at a time. The Hadamard gate has the important capacity of creating a superposition of the basis states from either one of the two basis states. This can be seen by letting it operate on our arbitrary qubit |αi,

H |αi .

= 1

√2

1 1

1 −1

A B

= 1

√2

A + B A − B

. (15)

Note that both A and B are present on both rows after the Hadamard gate has acted on the qubit. Another group of one-qubit gates, known as the phase shift gates, is given by

R(φ) .

=1 0 0 e^iφ

, (16)

which induces a phase shift of φ between the basis states. The Hadamard gate and the phase shift gates are together capable of simulating any unitary single-qubit opera-

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tion [28], which can be seen by letting the state |0i undergo the following transformation,

|0i → R (φ + π/2) HR(θ)H |0i .

= 1 2

1 0

0 e^i(φ+π/2)

1 1 1 −1

1 0 0 e^iθ

1 1 1 −1

1 0

= e^iθ/2

cos(θ/2) sin(θ/2)e^iφ

. (17)

Here, the overall phase factor e^iθ/2is unimportant and can be dropped. From gure 3 one can see that this transformation can create any possible qubit state. In this thesis only one-qubit gates are studied, however, to perform quantum computations it is necessary to also have a quantum gate that operates non-trivially on pairs of qubits. One such gate is the controlled-NOT gate which has two qubits, called control and target, as inputs.

The CNOT gate ips the target qubit if the control qubit is |1i while it does nothing if the the control qubit is |0i. In matrix representation this gate is written

U_CNOT .

=







1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0







, (18)

where the basis is {|00i , |01i , |10i , |11i} in that order, and with the control qubit written

rst. The Hadamard gate, the phase shift gates, and the controlled-NOT gate form a universal set of quantum gates [18, p. 22], that is, they are together able to simulate any quantum computation.

2.6. The density operator and open quantum systems

A quantum mechanical state that can be represented by a state vector (ket) is a pure state, i.e., operations on the ket predict the statistical outcome of measurements made on a pure ensemble (a collection of particles in identical states) [29, pp. 174-187]. But a state corresponding to a mixture of dierent pure ensembles cannot be described by a state vector. Such a collection of particles is called a mixed ensemble and the corresponding state is called a mixed state. A mixed state is characterised by the density operator

ρ =X

k

wk|α^(k)i hα^(k)| , (19)

where wk are the statistical weights of the pure states |α^(k)i. Thus, wk are non-negative numbers that satisfy wk ≤ 1and P_kw_k = 1.

To see how the density operator naturally arises, let us compute the expectation value of an observable A (i.e., A is a Hermitian operator) given a mixed ensemble consisting of N dierent pure ensembles with statistical weights wk. Naturally, the expectation value is obtained by calculating the expectation value for each pure ensemble and then computing their weighted sum. Thus, the expectation value of A in the mixed ensemble reads

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[A] =

N

X

i=1

w_ihα⁽ⁱ⁾|A|α⁽ⁱ⁾i , (20)

where the bracket is the usual expectation value of a pure state. By expanding equation (20) in the eigenstates {|bi} of another Hermitian operator B, the expectation value becomes

[A] =

N

X

i=1

X

b

w_ihα⁽ⁱ⁾|bi hb|A|α⁽ⁱ⁾i . (21) The two brackets are just complex numbers, thus, they commute and can be rearranged as

[A] =

N

X

i=1

X

b

wihb|A|α⁽ⁱ⁾i hα⁽ⁱ⁾|bi

=X

b

hb| A

N

X

i=1

wi|α⁽ⁱ⁾i hα⁽ⁱ⁾|

!

|bi . (22)

Here, the density operator in equation (19) has clearly emerged, and one obtains [A] =X

b

hb|Aρ|bi

= tr (Aρ) . (23)

Thus, the expectation value of any observable can be calculated by evaluating A and ρ in any convenient basis, which is a fact that makes equation (23) very powerful. Further- more, the density operator is Hermitian and it has unit trace.

A closed quantum system is assumed to be isolated from the environment and therefore undergoes unitary time evolution, however, in practice no physical system is perfectly closed. In this context an open system is a system that interacts with the outside world.

Basically, there are two main approaches to studying open quantum systems: the Hamil- tonian and the Markovian [30, pp. V-VIII]. The Hamiltonian approach aims to fully describe the system and the environment in terms of quantum mechanics, and their interaction is described through an explicit interaction Hamiltonian. In the Markovian approach, the environment is thought of as not completely known, or, as too compli- cated to be described in detail. Instead, the focus is on the eective dynamics induced on the system by the environment. Here, the Markovian approach is used, which, as the name implies, is based on Markovian dynamics. This means that the evolution of the system is assumed to depend only on the nearest past of the system, which is an approximation that is satised if the environment is large, in equilibrium, and interacts quicker internally than with the system. The time evolution of a Markovian open system is described by the Lindblad master equation [31] given by

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dρ dt = −i

~[H, ρ] − 1 2

n

X

k=1

Γ^†_kΓ_kρ + ρΓ^†_kΓ_k− 2Γ_kρΓ^†_k , (24) where H is the Hamiltonian of the closed system and the Lindblad operators Γk correspond to dierent decoherence processes. Without the sum, equation (24) becomes the Liouville-von Neumann equation, which is the mixed state equivalent of the Schrödinger equation.

The formalism of Completely Positive Maps (CPM) can be used to describe the dierent evolution modes a Markovian open system can undergo. The CPM considered here is denoted by E, and it maps the space of density operators onto itself. A short description of the properties of CPMs is given by Carollo [32] and a more comprehensive coverage can be found in textbooks, for example [30, pp. 157-162] and [33]. Here, I settle with stating that it can be shown [30, pp. 140-142] that a CPM E always can be written as

ρ → ρ⁰ = E(ρ) =X

k

W_kρW_k^†, (25)

where Wkare Kraus operators [33] satisfying the trace preserving condition P_kW_k^†W_k= 1. Each Wk corresponds to a physical process that brings the system to the state ρ⁰ = W_kρW_k^†, and that occurs with probability p_k = tr W_kρW_k^†

. For small time intervals

∆t, equation (24) can be approximated by

ρ(t + ∆t) ≈ ρ(t) −i∆t

~ [H, ρ(t)] −∆t 2

n

X

k=1

Γ^†_kΓ_kρ(t) + ρ(t)Γ^†_kΓ_k− 2Γ_kρ(t)Γ^†_k . (26)

By dening [34] the Kraus operators as

W₀= 1 − i∆t

~ H⁰ (27)

and

W_k=√

∆tΓ_k, (28)

where

H⁰ = H −i~

2 X

k

Γ^†_kΓ_k (29)

is the non-Hermitian eective Hamiltonian of the system, equation (26) can be written as

ρ(t + ∆t) ≈ E ρ(t) =

n

X

k=0

W_kρ(t)W_k^†, (30)

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where the error is O(∆t²). The operator W0 corresponds to continuous processes and is known as the no-jump operator, while the operators Wk are called jump operators because they correspond to dierent discontinuous jumps. These discontinuous jumps can be, for example, the deexcitation from a state of higher energy to a state of lower energy.

An intuitive way of thinking about the time evolution of the system is to see each time step as a branching point where the time evolution of the system breaks up into n + 1 possible trajectories, each corresponding to a Kraus operator Wk. If no information is gained about which trajectory the system has taken, the time evolution results in an incoherent weighted sum of all possible trajectories, as stated by equation (30). In this case, the trajectory interpretation is just a nice way of thinking, but it lacks physical meaning. However, if measurements on the environment are performed in each time step, in such a way that information about the system is obtained, then the system can be forced to follow a single trajectory. This is the basic idea of the quantum jump approach used in this thesis.

Here, the considered trajectory is the one without discontinuous jumps, which corresponds to repeated operations with W0 on the state. To only follow this trajectory is an approximation that is valid under the assumption that the system is only weakly perturbed by its environment. For each time step a pure state changes as

|α, t0; ti → |α, t0; t + ∆ti = W0|α, t0; ti

= |α, t₀; ti − i∆t

~ H⁰|α, t₀; ti , (31) which in the continuous limit (∆t → 0) becomes

i~d

dt|α, t0; ti = H⁰|α, t0; ti . (32) This looks like a Schrödinger equation, but it is governed by the non-Hermitian Hamil- tonian H⁰.

2.7. Operator delity

Fidelity measures the similarity of two quantum states and is dened as

F (α₁, α₂) = |hα₁|α₂i| . (33) A delity of 1 means that |α1iand |α2irepresent the same physical state, while a delity of 0 means that they are orthogonal. Assume that |α1i = U1|α0i and |α2i = U2|α0i, i.e., the two states are evolved from the same initial state |α0i by two dierent time evolution operators U1 and U2, hence, equation (33) can be written as

F (α1, α2) =

hα0|U₁^†U2|α0i

. (34)

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In this thesis, it is the similarity between time evolution operators representing gate transformations that is of interest. One way to construct a delity measure for such operators is to take the average delity over many appropriately distributed input states, which is given by

[F] = lim

N →∞

1 N

N

X

k=1

hαk|U₁^†U₂|αki

. (35)

Assume that a two-level system (e.g., a spin-¹₂-system) is studied, and that N is an even number. Then, the input states can be chosen to be pairwise orthogonal by picking N/2 states from the upper hemisphere, and N/2 states from the lower hemisphere of the Bloch sphere. Equation (35) can then be written as

[F] = lim

N →∞

1 N

N/2

X

k=1

hα_k|U₁^†U2|α_ki +

hα^⊥_k|U₁^†U2|α^⊥_ki

, (36)

where each pair satises hα_k|α^⊥_ki = 0. The trace of a matrix is independent of the basis used and two orthogonal states constitute a complete basis for a spin-¹₂ system, so by using the triangle inequality the following relation is obtained

[F] ≥ lim

N →∞

1 N

N/2

X

k=1

hαk|U₁^†U₂|αki + hα^⊥_k|U₁^†U₂|α^⊥_ki

= lim

N →∞

1 N

N/2

X

k=1

trU₁^†U2

= 1 2

trU1^†U2

= F (U¹,U2). (37)

Thus, F (U1,U2)can be interpreted as a lower bound of the average delity. This measure was introduced under the name operator delity by Wang et al. [35].

3. Geometric quantum gates for spin-

¹₂

systems

Geometric quantum computation is based on Berry's geometric phase factor described in section 2.3. It arises when non-degenerate energy eigenstates adiabatically undergo a cyclic evolution driven by a parameter dependent Hamiltonian. The eigenstates generally pick up dierent geometric phases, which means that an arbitrary state, initially in a superposition of the two eigenstates, has been altered. Thus, an input state has been transformed into an output state, where the transformation solely depends on the geometry of the path taken in parameter space. This is the basic idea of geometric quantum computation. The gate transformation is described by the time evolution operator of the physical system that constitutes the gate. This operator satises |α, t0; ti = U (t, t0) |α, t0; t0i, where |α, t0; t0i is an arbitrary input state that has been transformed into the output

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state |α, t0; tiby the time evolution operator U (t, t0).

(a) t = 0 → t = t1 (b) t = t1→ t = t2

(c) t = t2→ t = t3 (d) t = t3→ t = t4

Figure 4: In (a) the black vector represents the initial direction of the magnetic eld, and the blue and the red vectors represent two states prepared in the eigenstates of the initial Hamiltonian. As time goes from 0 to t1, the magnetic eld rotates one revolution around the z-axis. Assuming adiabatic time evolution, the two states remain instantaneous eigenstates of the Hamiltonian, and their paths are indicated by the blue and red curves. In (b), a static magnetic eld along the y-axis is applied for the time it takes for the two states to rotate a half revolution. The direction of rotation is not important, however, this direction is for a particle with positive magnetic moment.

The second cyclic evolution is shown in (c), which is in the opposite direction compared to (a). In the last step (d), the blue and red states are changed back, i.e., at time t4 the states are back to their initial positions as shown in (a).

The type of system considered here uses qubits consisting of spin-¹₂ particles, and the computational basis consists of the eigenstates of the spin operator ˆn · S, where ˆn is some chosen direction in the xed laboratory frame. The system is evolved in time by a magnetic eld with constant magnitude that rotates one revolution around the z-axis while keeping the polar angle xed, as seen in gure 4a. However, a state that is evolved in time picks up an eigenenergy-dependent dynamical phase factor in addition to the geometric one. Since the eigenenergies of the two spin-¹₂ eigenstates are split in the pres- ence of a magnetic eld (Zeeman splitting), the eigenstates do not generally obtain the

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same dynamical phase contribution, i.e., the relative phase is changed by the dynamical phases. Thus, to construct a purely geometric gate the dynamical phase factors must be either eliminated, or calculated and compensated for. In their NMR setup, Jones et al. [21] eliminated the dynamical phases by using a refocusing technique known as spin echo [28], and the same method is used in this thesis. The idea is to perform two cyclic evolutions in such a way that the second cyclic evolution cancels out the dynamical phases of the rst while the geometric phases are piled up. This is accomplished by letting the second cyclic evolution retrace the rst, but in the opposite direction, and by swapping the instantaneous eigenstates before and after the second cyclic evolution.

In practice, the swap operation can be performed by applying a static magnetic eld perpendicular to the direction of the nal (and initial) rotating magnetic eld for exactly the time it takes for a spin state to precess a half revolution around the direction of the static magnet eld. Thus, the time evolution of the qubit is obtained by solving the Schrödinger equation for four dierent time segments:

(i) In the rst step the magnetic eld rotates one revolution around the z-axis with constant magnitude of the magnetic eld and with xed polar angle. The path is visualised in gure 4a. Due to spherical symmetry, the initial magnetic eld vector can be chosen to lie in the xz-plane without loss of generality. The rotation starts at time t = 0 and has completed one lap at time t = t1 = T. Thus, for t ∈ [0, T ] the magnetic eld is described by

B(t) = Bsin(θ0) cos ϕ(t)ˆx + sin(θ₀) sin ϕ(t)ˆy + cos(θ₀)ˆz , (38) where B is the magnitude of the magnetic eld, θ0is the xed polar angle, and ϕ is the azimuthal angle satisfying ϕ(0) = 0 and ϕ(T ) = 2π. The time evolution of the system is governed by the Hamiltonian H(t) = −µ · B(t), where µ = ^g_2m^s^eS is the spin magnetic moment of the particle expressed in SI-units. Here, e is the charge of the particle (e < 0 for an electron), gs is the spin g-factor, m is the particle's mass, and S is the spin vector. By introducing the Larmor frequency ω0 = −^g_2m^s^eB, which is the rate of spin precession around the magnetic eld, the Hamiltonian can be written as

H(t) = ω0B(t) · S,ˆ (39)

where ˆB(t) is the unit vector of the magnetic eld. By using the rotation operators Dz(φ) = exp_−iφS

z

~

and Dy(φ) = exp_−iφS

y

~ , the Hamiltonian can be parametrised as

H(t) = ω0Dz ϕ(t)H0Dz⁻¹ ϕ(t). (40) Here, H0 = H(0)/ω0, which in parametrised form becomes

Fidelity of geometric and holonomic quantum gates for spin systems

Examensarbete 30 hp April 2014

Fidelity of geometric and holonomic quantum gates for spin systems

Daniel Töyrä

Abstract

Fidelity of geometric and holonomic quantum gates for spin systems

Populärvetenskaplig sammanfattning

Contents

List of Figures

1. Introduction

2. Notations and important concepts

3. Geometric quantum gates for spin-

systems