Quantum Speed Limits of Spin Chain Dynamics: Applying Quantum Speed Limits to Spin Chains used as Quantum Channels

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Quantum Speed Limits of Spin Chain Dynamics

Applying Quantum Speed Limits to Spin Chains used as Quantum Channels

Abstract

Quantum computers, like classical ones, need to be capable of sending information between different parts of the computer. Spin chains are one viable method of sending qubits while preserving the superposition and entanglement of a state. We use quantum speed limits, which give bounds on how fast non-relativistic quantum systems can evolve, to study simple Heisenberg chains and their efficacy as quantum channels.

Project in Physics and Astronomy Author: Yonatan Gazit

Supervisor: Gregor K¨ alin

Department of Physics and Astronomy Uppsala University

June 3, 2017

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1 Introduction

Since the mid-20^thcentury, computational power has doubled roughly every 18 months.

This trend in computational progress is called Moore’s Law [14], and has been ac- complished through new technologies to make information processing components both smaller and cheaper to manufacture. There are physical limits to how far technology can progress, though. If computers continued to follow Moore’s Law for the next 20 years, computers would be preforming operations on the scale of individual atoms [11].

Quantum Computing tries do to exactly this: embed information in quantum systems and manipulate them to preform computations. It not only allows for computation to be carried out on scales otherwise impossible due to quantum effects, but it has led to new proposals for types of computation using the properties of quantum mechanics.

One of the best known examples of this is Shor’s Algorithm [21]. It carries out prime factorization in polynomial time, as opposed to a classical computer, where the best algorithm known does so in exponential time [17]. There is a class of computational problems called NP which have no quick solution on a classical computer, but solutions can be checked quickly on a classical computer. Prime factorization is an example of an NP problem which is quickly solvable on a quantum computer. Another such example is the quantum search algorithm of an unordered list, Grover’s algorithm [7]. One aspiration of quantum computing is to provide more solutions to such NP problems.

Experimental realization of quantum computers is slow, due to the difficulty of as- sembling and purposefully manipulating large quantum systems. Quantum channels that transfer information could reduce the complexity of building a quantum computer. The size of individual quantum systems can be kept only as large as they need to be for carrying out specific computations. However, such a channel is not as simple as sending an electrical signal encoding a series of bits. Channels also must preserve the entanglement and superposition of states – topics which will be covered in more depth in section 2.1.

The time it takes for the information to travel through a quantum channel is also an important factor of a channel.

Spin chains are a viable candidate for a quantum channel over small distances, such as within a quantum computer. For larger distances, photons are usually preferred due to

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their speed. A variety of spin chains have been proposed as efficient quantum channels [3].

However, given the differences in and complexities of the spin chains proposed, comparing their speeds at transferring quantum information and trying to optimize each chain indi- vidually is challenging. Quantum speed limits, relations which give upper bounds on how quickly a state can change in non-relativistic quantum mechanics, offer a straightforward way to do such comparisons and analyses.

This report will begin with an introduction to a few concepts in Quantum Information Theory to provide the reader with the required background to understand the main body of this report. It is assumed readers have the background knowledge of an introduction to quantum mechanics course. Then, quantum speed limits will be introduced and derived.

The report will go on to introduce spin chains and how they are used as quantum channels.

We will use the previously derived quantum speed limits to analyze how spin chains may be optimized as quantum channels. A scheme for perfect state transfer using parallel spin chains will be introduced, and a second scheme using engineered spin chains will be reviewed.

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2 Building Blocks of Quantum Computation

A few basic concepts from quantum information theory must first be covered to understand how a quantum system can act as a quantum channel. First we will introduce qubits, the basic information-carrying unit of quantum computation. In quantum computation it is often necessary to work with a statistical ensemble of states as opposed to one definite state. Quantum channels are no exception to this, and so density operators will also be covered. Lastly, the fidelity between quantum states will be introduced as a way to measure how effective a quantum channel is.

2.1 Qubits

A bit is one of the building blocks of classical computation. It is represented as a 1 or 0, physically corresponding to a small circuit with voltage above or below a threshold value. In quantum computation, the qubit (quantum bit) plays an analogous fundamental role. One of the postulates of quantum mechanics is that all quantum systems occupy a Hilbert space [1]. A qubit is any quantum mechanical system in a two-dimensional Hilbert space with two possible states, |0i and |1i. The main difference between a bit and a qubit is that a bit is always exclusively in either a 0 or 1 state, but a qubit can be in a superposition of its two states

|ψi = α|0i + β|0i, (2.1)

where α, β ∈ C and |α|² + |β|² = 1. The second condition is necessary due to the wave function representing a statistical distribution. Measurements of a the state in eq. (2.1) would yield the state |0i with probability |α|² and |1i with probability |β|². For the wave function to represent a valid probability distribution, it must be normalized.

Qubits have been experimentally realized in different quantum systems, such as the polarizations of a photon, the alignment of nuclear spins in a uniform magnetic field, and the excitation states of an electron [17], to name a few. For the purposes of this report, though, a qubit will be thought of as any quantum mechanical system that has a |0i and

|1i state. For a spin chain, this manifests itself in the quantized spins coupled to one

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Figure 1: Bloch sphere representation of a qubit [17].

another along a chain.

We can rewrite the coefficients for the qubit as |ψi = eîγ cos(^θ₂)|0i + eîφsin(^θ₂)|1i, where γ, φ, and θ are all real numbers, as the coefficients represent a normalized wave function. The eîγintroduces an overall phase difference to the qubit and has no observable effects, so this term can be ignored. Thus, the expression can be written as

|ψi = cosθ

2|0i + e^iφsinθ

2|1i. (2.2)

The variables θ and φ determine a point on a 2-dimensional sphere, called a Bloch Sphere.

Eq. (2.2) is called a Bloch sphere representation of a qubit, shown geometrically in figure 1.

The Bloch sphere representation does not offer new insights about qubits, but it does help build a geometric intuition for what operations do to a qubit. In classical computers, operations on a bit are represented as a series of logic gates. For example, a NOT gate will flip a bit to its opposite state. A similar analogy can be used for operations on qubits.

To understand what operations do to a qubit, it is easier to first look at the qubit as a vector, expressed in the orthonormal basis |0i and |1i, where

|0i =



 1 0



, |1i =



 0 1



. (2.3)

The vector |ψi = α|0i + β|0i expressed in this basis is

|ψi =



 α β



. (2.4)

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For a qubit, the equivalent of a NOT gate is the Pauli spin matrix along the x-axis σ_x written here as just X

X|ψi =



 0 1 1 0







 α β



=



 β α



. (2.5)

For the special case where α or β = 1, then the X gate acts exactly as a classical NOT gate would. On the Bloch sphere this operation is equivalent to a rotation around the x-axis by π.

The controlled-NOT gate (CNOT) is another gate used in quantum computation, and it appears in a protocol outlined in section 5.5. The gate has two input qubits, a control and target. The CNOT gate applies a NOT gate to the target qubit only if the control qubit is in the state |1i. To illustrate with an example, take a control and target qubit initialized in the states |φ_ci = α|0i + β|1i and |φ_ti = |1i. After applying the CNOT gate,

|φci will be unchanged, but the target qubit will be in the state |φti = β|0i + α|1i. The CNOT gate transforms the target qubit into the state equivalent to a NOT gate applied to the control qubit.

2.1.1 Entanglement of Qubits

A system comprised of two wave functions from their respective Hilbert spaces |ψ₁i ∈ H₁ and |ψ₂i ∈ H₂ is represented by the tensor product of these two states, |ψ₁i ⊗ |ψ₂i ∈ H₁⊗ H₂ [1]. If H₁ has complex dimension n and H₂ has complex dimension m, then their tensor product H₁⊗H₂ has complex dimension nm. A n-particle system is represented by the tensor product of all the composite systems, |ψ₁i⊗|ψ₂i⊗· · ·⊗|ψ_ni ∈ H₁⊗H₂⊗· · ·⊗H_n. This is also written as |ψ₁ψ₂. . . ψ_ni.

Each qubit occupies a 2-dimensional Hilbert space, and an n-qubit system occupies a 2ⁿ-dimensional Hilbert space. This exponential growth of the Hilbert space with size is part of the challenges in building a large scale quantum computer. The larger the quantum system, the more susceptible it is to environmental noise. However, this growth also gives rise to new possibilities with quantum computers.

Simulating large quantum systems is another promise of quantum computation [10]. If

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each particle of an n-particle system occupies two-dimensional Hilbert space, a classical computer must keep track of 2ⁿ variables, or through simplifications slightly fewer, to simulate it. Still, there is an exponential increase in computational requirements as the size of a system increases. A quantum computer could manipulate ensembles of qubits to simulate a specific quantum system. This requires only a polynomial increase in amount of computational resources as the system grows. Another consequence of this feature is that quantum computers would only need to grow by one qubit every 18 months to keep up with Moore’s law [17]. Adding a single qubit to a quantum computer corresponds to a doubling of computation ability in classical computer.

Entanglement is a feature found only in quantum systems, and it leads to new possibilities with quantum computers like quantum teleportation (see 1.3.7 in [17] for more information). Entanglement will be explained here through the Bell states, the group of states

|β₀₀i = |00i + |11i

√2 |β₀₁i = |01i + |10i

√2

|β10i = |00i − |11i

√2 |β11i = |01i − |10i

√2

(2.6)

which are maximally entangled two-qubit states. They are used in quantum teleportation and other protocols that take advantage of quantum entanglement. We define entanglement as a system that can not be written as a product of its component systems [17], for example there are no states |ψi and |φi such that |β₀₀i = |ψi⊗|φi. None of the Bell states can be written as a product of component states. For any multi-particle system with this feature, its components are entangled. Entanglement does not exist in classical systems.

A classical system comprised of other systems can always expressed as a combination of smaller systems.

2.2 Density Operators

In Quantum Information Theory, the exact state of a system is not always known.

After a quantum operation, we might only know the probability of our qubit being in a certain state, or after considering the noise in a system we might be able to say that only up to some probability a qubit is in the desired state. To account for these uncertainties a system is described by a density operator, a probability distribution over a set of quantum

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states.

2.2.1 Definition and Properties of the Density Operator

If a system is known to be in a range of states, |ψ_ii each with probability p_i, it is represented by an ensemble of pure states, {p_i, |ψ_ii}. From this ensemble, a density operator is constructed

ρ ≡X

i

p_i|ψ_iihψ_i|. (2.7)

The ensemble of states must satisfy the condition that P

ip_i = 1 in order to be a valid probability distribution. This property helps us define the basic properties of any density operator [19],

ρ 0 (2.8)

Tr(ρ) = 1. (2.9)

Density operators can be thought of as either being an ensemble of pure states, or operators that adhere to eq. (2.8) and (2.9). Density operators of the form ρ = |ψihψ| are called pure states, since they correspond to a single known state. Otherwise, they are called mixed states. A pure state satisfies Tr(ρ²) = 1. The completely mixed state has the form ρ = ¹_dI_d, in which there is an equal probability of all possible states.

If a closed system undergoes a unitary transformation described by an operator U, the evolution of the system written in terms of the density operator is

ρ =X

i

p_i|ψ_iihψ_i|−→^U X

i

p_iU |ψ_iihψ_i|U^† = U ρU^†. (2.10)

Returning to the Bloch sphere representation of a qubit

|ψi =



 cos^θ₂ e^iϕsin^θ₂



, (2.11)

we write the Bloch representation of a density operator in a pure state through a transformation to spherical coordinates. If r = 1, we have the relation between the coordinates

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to be x = cos ϕ sin θ, y = sin ϕ sin θ, z = cos θ.

|ψihψ| =



 cos^θ₂ e^iϕsin^θ₂





cos^θ₂ e^−iϕsin^θ₂

=





cos^{2 θ}₂ e^−iϕcos^θ₂sin^θ₂ e^iϕcos^θ₂sin^θ₂ sin^{2 θ}₂





=





1+cos θ

2 (cos ϕ − i sin ϕ)^{sin θ}₂ (cos ϕ + i sin ϕ)^{sin θ}₂ ^{1−cos θ}₂





= 1 2





1 + z x − iy x + iy 1 − z





= 1 2



 1 0 0 1



+



 0 x x 0



+





0 −iy iy 0



+





z 0

0 −z





= I + ~r · ~σ

2 , (2.12)

where ~σ = (σx, σy, σz), the Pauli matrices, and ~r = (x, y, z). Eq. (2.12) is called the Bloch Vector of a density operator. The Bloch vector of a mixed state is a summation of the Bloch vectors of its ensemble states multiplied by their probability pi. Note that ||~r|| ≤ 1, while only equal to one in the case where ρ is a pure state.

2.2.2 Partial Trace and Reduced Density Operators

When dealing with multi-particle systems, often just a specific part of the system is of interest, such as the receiving end of a spin chain. The partial trace is used to recover specific parts of a composite system [19]. Given two Hilbert spaces H_A and H_B, with

|a_ii, |b_ii being vectors in those spaces, the partial trace is defined to be

Tr_B(|a₁iha₂| ⊗ |b₁ihb₂|) ≡ |a₁iha₂| Tr(|b₁ihb₂|) (2.13)

If we have a system

ρ^AB = ρ^A⊗ ρ^B, (2.14)

where ρÂB is a composite system made out of ρÂ and ρ^B. We recover ρÂ, the reduced density operator, using the partial trace.

ρ^A= Tr_B(ρ^AB) (2.15)

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If we know the Eigenbasis of H_B, taken here to be |i_Bi , then the partial trace over B is

ρ^A = Tr_B(ρ^AB) =X

i

(I ⊗ hi_B|)ρ^AB(I ⊗ |i_Bi). (2.16)

The Schmidt Decomposition of a joint system |Ψi ∈ H_A ⊗ H_B lets us express it as

|Ψi = P

iγ_iu_i ⊗ v_i. A density operator can then be defined, ρÂB = |ΨihΨ|, as well as and two reduced density operators ρÂ = Tr_B(ρÂB) and ρ^B = Tr_A(ρÂB). The orthonormal basis u_iconsists of the Eigenvectors of ρÂ, and v_iof ρ^B. The values γ_i² are the Eigenvalues of both ρÂ and ρ^B.

A purification on any mixed state ρ of a quantum system A is preformed using the partial trace. We introduce a reference system R such that |ARi is a pure state. The system R has no physical meaning, but for certain calculations it is easier to work with the purification of ρ instead of the density operator itself. This is the case with certain quantum speed limits. If ρ has a decomposition ρ =P

ipi|i^Aihi^A|, then R has the same state space as A with the basis states |i^Ri. The pure state of the combined system is defined as

|ARi ≡X

i

√p_i|i^Ai|i^Ri. (2.17)

Taking the partial trace over the reference system recovers the mixed density operator ρ.

2.3 Fidelity

Fidelity is a measure of how close two quantum states are to one another. The ability of quantum channel to accurately transfer a state is measured through the fidelity between the input state and the state recovered at the other end of the chain. It is defined as

F (ρ, σ) ≡ Trp

ρ^1/2σρ^1/2. (2.18)

The positive square root of an operator δ = √

ρ is any operator such that δδ = ρ.

The definition of density operator includes that it is positive-definite (eq. (2.8)), and a positive-definite matrix has only one square root. The density operator ρ has a spectral decomposition ρ = T DT⁻¹ where D is a diagonal matrix with the Eigenvalues of ρ and T is a matrix with the corresponding Eigenvectors. We find √

ρ by re-expressing

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the decomposition as ρ = (T D^1/2T⁻¹)(T D^1/2T⁻¹). Because D is a diagonal matrix, D^1/2 is a diagonal matrix of all the positive square roots of the Eigenvalues. Therefore, δ = (T D^1/2T⁻¹), and due to its uniqueness the fidelity is well defined.

The fidelity between two density operators undergoing the same dynamcis does not change over time

F (U ρU^†, U σU^†) = F (ρ, σ) (2.19) If we take the fidelity between a pure state γ = |ψihψ| and an arbitrary density operator ρ, we get

F (γ, ρ) = Tr p

γ^1/2ργ^1/2

= Tr p|ψihψ|ρ|ψihψ|

F (γ, ρ) =phψ|ρ|ψi, (2.20)

where the last step is attained from the fact that hψ|ρ|ψi after evaluation is a scalar value while |ψihψ| is a matrix of a pure state, so its trace is equal to 1. We interpret the fidelity to be the amount of overlap the density operator has with the state |ψi. If ρ is also a pure state, then the Fidelity is the same as taking the inner product F (φ, ψ) = |hφ|ψi|.

Considering the limiting cases when φ and ψ are parallel or orthogonal to one another, we see that 0 ≤ F (ρ, σ) ≤ 1.

|φ_ai. Another way of measuring how distant a density operator ρ is from a desired state ψ is through this fidelity-squared,

F²(ρ, ψ) = hψ|ρ|ψi (2.21)

Other measures of distance between density operators are used, such as statistical metrics [18] or the quantumness between two states [8]. However, for the case of spin chains, the fidelity describes in a straightforward manner the distance between a desired state and a density operator. This is sufficient for measuring how effective a spin chain

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is as a quantum channel.

3 Quantum Speed Limits

Quantum Speed Limits were first derived in the 1940s through time-energy uncertainties [12]. They set a bound on how fast a non-relativistic quantum system can evolve with time under unitary dynamics. When a system is big enough that its behavior over time cannot be found analytically or computationally, we can use quantum speed limits to analyze its time evolution.

3.1 Mandelstam-Tamm Limit

In 1945 Mandelstam and Tamm published the first quantum speed limit based on the energy-variance of the system in question [12]. One of the postulates of quantum mechanics puts forth a minimum uncertainty between the any two observables, ∆A∆B ≥

1

2h[A, B]i, where hAi is the expectation value of that operator, ∆B = phB²i − hBi², and [A, B] = AB − BA is the commutator between A and B. For a time independent Hamiltonian, the partial derivative of the expectation value of an observable is _∂t^∂hAi =

i

~h[A, H]i. Combining these two, the uncertainty between the variance in the energy and any observable A is

∆E∆A ≥ ~

∂hAi

∂t

, (3.1)

The Eigenvectors of A must satisfy Av = av. Because A is the projection onto the initial state |ψ₀i, the only possible Eigenvectors are states either parallel or orthogonal to |ψ₀i. This means that the Eigenvalues of A are either 1 or 0, so A² = A. The variance is then simplified to ∆A = phA²i − hAi² = phAi − hAi², which allows eq. (3.1) to be

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rewritten as

∆EphAi − hAi² ≥ ~ 2

∂

∂thAi

(3.2) Our focus will now be to integrate this differential equation. For clarity in the following calculation hAi will be replaced with x. We can then rewrite the terms as

∆E√

x − x² ≥ ~ 2

∂

∂tx

∆E

~ ≥ 1

2√ x

√ 1 1 − x

∂

∂tx

(3.3)

The right hand side of eq. (3.3) looks like a total time derivative of a function f (x),

d

dtf (x) = _dx^df (x)_∂t^∂x. Since the absolute value of the partial derivative is taken, we can choose f (x) to be either positive or negative with respect to t. The range of x is in x ∈ [0, 1] because hAi is the fidelity-squared. We integrate everything with respect to time and introduce the substitution u =√

x to get Z ∆E

~ dt ≥

Z 1

2√ x

√ 1 1 − x

∂

∂tx

dt,

⇐⇒ ∆E · t

~ ≥

Z −1 2√

x

√ 1

1 − xdx,

=

Z −1

√1 − u²du,

⇐⇒ ∆E · t

~ ≥ arccos

phAi(t)

,

hAi(t) ≥ cos² ∆E · t

~

. (3.4)

The time it takes for |ψ0i to evolve to an orthogonal state is the same as the time it takes to go from hAi(0) = 1 to hAi(τ ) = 0, for a later time τ . From eq. (3.4), we get the relation that the minimum time τ it takes to evolve to an orthogonal state is

τ ≥ π~

2∆E (3.5)

Note that this quantum speed limit only applies to unitary dynamics and time independent Hamiltonians.

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3.2 Margolus-Levitin Limit

One criticism of the quantum speed limit in eq. (3.5) is that it implies that any system can be sped up indefinitely by increasing the energy variance [6]. Another bound dependent only on the average energy of the system E = hψ|H|ψi was developed in [13].

Starting from an initial state |ψ_ii described in its energy decomposition |ψ_ii =P

nc_n|E_ni, the system will evolve with time to the state

|ψ_ti =X

n

c_ne^−iEⁿ^t/~|E_ni (3.6)

The fidelity-squared changing over time between these two states F²(t) = hψ_i|ψ_ti written in the energy Eigenbasis is

F²(t) =X

n

|c_n|²e^−iEⁿ^t/~ (3.7)

The trigonometric inequality cos x ≥ 1 −²_π(x + sin x) when x ≥ 0 is the key to this mean energy bound. Taking the real part of F²(t),

Re(F²(t)) = X

n

|c_n|²cos(^Eⁿ^t

~ ), (3.8)

≥X

n

|c_n|²

1 − π

2

_E_n_t

~ + sin(^Eⁿ^t

~ )

, (3.9)

= 1 − 2E π~t + 2

πIm(F²(t)). (3.10)

F²(t) = 0 when ψ_t is orthogonal to ψ_i, so it follows that Re(F²(t)) = Im(F²(t)) = 0.

Eq. (3.10) then becomes

0 ≥ 1 − 2E

π~t, (3.11)

or rewritten to give a time bound, the minimum time τ it takes to evolve into an orthogonal state is

τ ≥ π~

2E. (3.12)

This quantum speed limit was found 50 years after Mandelstam and Tamm published theirs.

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3.3 Comprehensive Energy Quantum Speed Limit and Other QSLs

Given two different bounds for the amount of time it takes a quantum system to evolve to an orthogonal state, this comprehensive energy quantum speed limit is conveniently written as

τ ≥ max

π~

2∆E, π~

2E

, (3.13)

where E = hψ|H|ψi, and ∆E =phψ|(H − E)²|ψi. Even though both proofs give a valid bound for the time needed for a state to evolve to an orthogonal state, the maximum between the two gives the tighter bound and thus better limit.

Although eq. (3.13) assumes time-independent Hamiltonians, other derivations have been done for time-dependent ones to give similar bounds [2]. Other work has been done to extend these quantum speed limits to evolution between non-orthogonal states [6] and non-unitary dynamics [5].

Quantum speed limits have also been developed from a geometric point of view [18].

Because density operators are just statistical ensembles, a metric on the space of density operators may be defined and the group of all density operators can be treated like a Riemann manifold. Then, a relation for the evolution of a density operator is defined as movement through the manifold, and a geodesic is found to describe the shortest path and subsequently time an evolution takes.

One issue with the geometric approach is that the geodesic and subsequent quantum speed limit depend on the metric used. There are different ways of defining the difference between two statistical ensembles. In a 1-variable case, the mean, standard deviation, and kurtosis of a distribution must be considered when defining how different one distribution is from another, leading to more than one possible metric. However, the strengths of a geometric approach come from its focus on the density operator and its initial and final states. This way, it better handles non-unitary dynamics compared to the quantum speed limit defined in eq. (3.13).

A recently proposed quantum speed limit is based on the quantumness of in initial and final states [8]. This quantumness measure between two states is defined as how

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much the initial and final state commute with one another. Its strengths, like those of the geometric quantum speed limits, come from the fact that such a focus on the initial and final state makes it quite useful to describe non-unitary dynamics.

For the purposes of spin chains, though, the energy quantum speed limit is enough to provide insight into the dynamics of the system.

4 Spin Chains as Quantum Channels

A series of coupled spin-1/2 systems, forming a spin chain, can act as a quantum channel to transfer the state of a qubit from one part of a quantum computer to another.

The superposition of a qubit then propagates through the spin chain via the interactions between each spin and is recovered at the other end of the spin chain.

4.1 Introduction to the Heisenberg Spin Chain

An interaction between two spins is written as

H_ij = J_ijS~_i⊗ ~S_j, (4.1)

where ~S_i⊗ ~S_j ≡ S_i^xS_j^x+S_i^yS_j^y+S_i^zS_j^z. This is called a Heisenberg Interaction. For spin-1/2 systems, ~S are the Pauli matrices. A series of spins coupled as in eq. (4.1) is called a XXX Heisenberg Chain, where the three Xs signify that coupling is the same in all three spatial dimensions. A XXZ chain would signify that coupling along the z-axis is different from the other two. A common variant of the of spin-spin interaction in eq. (4.1) is

H_ij^XY = J_ij(S_i^xS_j^x+ S_i^yS_j^y). (4.2)

Spin chains coupled according to eq. (4.2) are referred to as XX Heisenberg Chains.

To transfer a quantum state, we take J_ij < 0. This describes a ferromagnet, where it is energetically favourable for all spins to be facing the same direction. Such spin systems lead to a useful understanding of the information moving through the chain as spin waves,

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which will be covered in section 5.1. We will use the fidelity-squared between the input and output states (eq. (2.21)) to measure how well our spin chain transfers a quantum state, referred to here as just F (ρ, σ).

The Hamiltonian of a spin chain with N spins is given as

H = −X

hi,ji

J_ij~σⁱ~σ^j−

N

X

i=1

B_iσ_zⁱ, (4.3)

where ~σ are the Pauli matrices, B_i is an external magnetic field, and < i, j > describes a summation over neighboring interactions. To describe a ferromagnet, both B_i > 0 and J_ij > 0. For the rest of this report, the quantum information we wish to send through a spin chain will be assumed to propagate along the z-axis.

4.2 Transferring a State in a General Spin Chain

As with qubits, the states of the spins along the spin chain can either be in the ground state, |0i, or in an excited, spin-flipped state |1i. The spin chain is initialized with all the spins in their ground state, referred to as |0i = |000 . . . 0i. For simplicity later on we define the ground state energy to be E₀ = 0, and redefine H = E₀+ H. To define our notation, we signify that a single spin of the chain flipped to the |1i state at the j-th site is |ji = |00 . . . 1 . . . 0i.

The sender will have an input state

|ψini = α|0i + β|1i = cosθ

2|0i + e^iφsinθ

2|1i, (4.4)

using the Bloch sphere representation of our input state from eq. (2.2). This input state is attach to the spin chain and becomes the first site. A spin flipped at the first site is signified as |si instead of |1i to avoid confusion with the general spin-flipped state |1i.

At a time t = t₀ the receiver will preform a measurement, a type of operation, or try to isolate the qubit at the N -th site of the spin chain in a state ρ_out. A spin flipped at the receiving qubit is referred to as |ri. At time t = 0 spin chain is in the state

|Ψ(t = 0)i = cosθ

2|0i + e^iφsinθ

2|si (4.5)

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Before discussing the evolution of the spin chain, the Transition Probability Amplitude will be introduced. For a system described by the Hamiltonian H initially in the state

|ψ_ii, the probability that it will later be in the state |φi is given by

P(t) = hφ|U_H|ψ_ii, (4.6)

where U = e^−iHt/~is the unitary time evolution of the system. Eq. (4.6) is also the fidelity- squared between the evolving state U |ψ_ii and |φi. This yields a probability distribution that varies with time. If the system |ψ_ii at a later time evolves into |φi, then P = 1.

If |ψ_ii evolves to a state orthogonal to |φi then P = 0. And at any other state in the evolution of |ψ_ii describes how much overlap there is between the two states.

In order to transfer the input qubit, we want |Ψ(t)i to evolve as close as possible to the state cos(θ/2)|0i + e^iφsin(θ/2)|ri. One way to describe this time evolution is through the general unitary transformation, U = e^−iHt/~. However, some simplifications can be made.

The spin chain is initially in the |0i state, so the contribution of the α part of our input state eq. (4.4) is automatically included in the receiving site. This is because we are transmitting a superposition of a |0i and |1i state, and our entire spin chain is in the ground state |0i. To properly transmit the superposition of the input state, the spin chain needs to transmit how much the |1i state made up the superposition of the input qubit. Thus, our main interest is to see how the β or e^iφsin(θ/2) part of the superposition in eq. (4.4) gets transmitted through the spin chain. We can then describe the evolution of the spin chain according to how much of the |1i part of the superposition is at each site, given by the transition probability amplitude

P_j(t) = hj|e^iHt|si. (4.7)

The value P_j(t) describes the transition probability amplitude for the j-th site of the spin chain. We see that a total sum over all sites must be equal to e^iφsin(^θ₂) of the input qubit

N

X

j=1

e^iφsinθ

2hj|e^iHt|si = e^iφsinθ

2. (4.8)

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This is because the chain was initially all in the ground state |0i, then due to unitary evolution of the system there could be no more of the excited |1i state in the spin chain than what was in our input qubit. The evolution of the spin chain is then

|Ψ(t)i = cosθ

2|0i + e^iφsinθ 2

N

X

j=1

P_j(t)|ji. (4.9)

We can then find the state of the r-th spin by first turning |Ψ(t)i into a density operator,

ρ(t) = |Ψ(t)ihΨ(t)| (4.10)

=

cosθ

2|0i + e^iφsinθ 2

N

X

j=1

P_j(t)|ji

cosθ

2h0| + e^−iφsinθ 2

N

X

j⁰=1

P_j^∗0(t)hj⁰|

ρ(t) = cos² θ

2|0ih0| + cosθ

2e^−iφsinθ 2

N

X

j⁰=1

P_j^∗⁰(t)|0ihj⁰|

+ cosθ

2e^iφsinθ 2

N

X

j=1

P_j(t)|jih0| + sin² θ 2

N

X

j=1 N

X

j⁰=1

P_j(t)P_j^∗0(t)|jihj⁰| (4.11)

Then, we can take the partial trace of ρ(t) to find the dynamics at the receiving site. First, let us take take the partial trace of just the first spin of this system. The orthonormal basis of any subsystem consists of |0i and |1i. The partial trace over the first site is

X

i=0,1

hi ⊗ I_{N −1}|ρ(t)|i ⊗ I_{N −1}i =

X

i=0,1

hi ⊗ I_{N −1}| cos² θ

2|0ih0| + cosθ

2e^−iφsinθ 2

N

X

j⁰=1

P_j^∗0|0ihj⁰|

+ cosθ

2e^iφsinθ 2

N

X

j=1

P_j|jih0| + sin² θ 2

N

X

j=1 N

X

j⁰=1

P_jP_j^∗0|jihj⁰|i ⊗ I_{N −1}i,

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= cos² θ

2|0^N−1ih0^N−1| + cosθ

2e^−iφsinθ 2

N

X

j⁰=2

P_j^∗0N −1|0^N−1ihj^0N−1|

+ cosθ

2e^iφsinθ 2

N

X

j=2

Pj|j^N−1ih0^N−1| + |P1|²sin² θ

2|0^N−1ih0^N−1| + sin² θ

2

N

X

j=2 N

X

j⁰=2

P_jP_j^∗0|j^N−1ihj^0N−1| (4.12)

Each partial trace for the sites 2, . . . , N − 1 will reduce summation indices by one and will add another |Pj|² term. The result is

ρ_out(t) = cos² θ

2|0ih0| + cosθ

2e^−iφsinθ

2P_r^∗(t)|0ih1| + cosθ

2e^iφsinθ

2P_r(t)|1ih0|

+ sin² θ

2|Pr(t)|²|1ih1| +

N −1

X

j=1

|Pj(t)|²sin² θ

2|0ih0| (4.13)

We can write this in more succinct form using the relation PN −1

j=1 P_j = 1 − P_N. This allows us to rewrite PN −1

j=1 |P_j|²sin^{2 θ}₂ = 1 − (cos^{2 θ}₂ + |P_N|²sin^{2 θ}₂), and the state of the receiving qubit as

ρ_out(t) = L(t)|ψ_out(t)ihψ_out(t)| + (1 − L(t))|0ih0|, (4.14)

|ψout(t)i = 1

pL(t) cos^θ₂|0i + e^iφsin^θ₂Pr(t)|1i, (4.15)

where L(t) = cos^{2 θ}₂ + sin^{2 θ}₂|Pr(t)|². If the last spin is isolated at time t = t0, then the average fidelity of the spin chain is computed by averaging over all input states:

Ftot = 1 4π

Z

hψin|ρout(t0)|ψinidΩ (4.16)

Taking the fidelity-squared between these states, we get that

hψ_in|ρ_out(t₀)|ψ_ini = cos²(^θ₂) + sin⁴(^θ₂)|P_r|²+ cos²(^θ₂) sin²(^θ₂) Re(P_r) − |P_r|²

(4.17)

In the complex plane Re(Pr) is found by Re(Pr) = |Pr| cos[arg(Pr)]. Plugging this all into the integral, we get that

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F_tot = 1 4π

Z 2π 0

dϕ Z π

0

dθ sin(θ)hψ_in|ρ_out(t₀)|ψ_ini (4.18)

= 1 2

Z π 0

dθ sin(θ)

cos²(^θ₂) + sin⁴(^θ₂)|P_r|² + cos²(^θ₂) sin²(^θ₂) cos[arg(P_r)] − |P_r|²

(4.19)

This ends up just being equal to

F_tot = |P_r| cos[arg(P_r)]

3 +|P_r|² 6 + 1

2 (4.20)

In this section we have built an understanding of how Heisenberg spin chains can be used as a quantum channel. The evolution of a spin chain is described entirely by the transition probability amplitudes, and the average fidelity of the spin chain was found to depend only on the transition probability amplitude of the initial and final sights.

5 Spin Chains and Quantum Speed Limits

With a general understanding of how a spin chain acts as a quantum channel, specific spin chains may now be analyzed to see how effective of a quantum channel they are.

This is done through both analytical methods and the quantum speed limits introduced in section 3.

5.1 Nearest Neighbor Spin Chain

The Hilbert space of an N -length spin chain is 2^N. For a N = 50 spin chain, the Hilbert space is around 112, 590, 000, 000, 000-dimensional. This makes any computation in such a large basis cumbersome and for too large cases impossible. Analytical diagonalizations, or even approximate analytical diagonalizations, offer a fast way to look at the behaviour of a spin chain and its efficacy as a quantum channel. The Hamiltonian of a spin chain with uniform external field B_i = B and nearest-neighbour interactions, J_ij = J δ_i,i+1 is analytically diagonalizable.

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The Hamiltonian of this chain is written as

H_XXX = −J

N

X

j=1

(σ_j^xσ^x_j+1+ σ_j^yσ_j+1^y + σ^z_jσ_j+1^z ) − B

N

X

j=1

σ_j^z (5.1)

The analytical solution for the diagonalization of this Hamiltonian is found via the Bethe Ansatz [16]. To understand how the Ansatz works, a review the spin operators σ^z, σ⁺, σ⁻ is helpful. Here, σ^± = σ^x± σ^y is the raising/lowering operator. Acting on a spin, they produce the result

| ↑ i | ↓ i σ^z ¹₂| ↑ i −¹₂| ↓ i

σ⁺ 0 | ↑ i

σ⁻ | ↓ i 0

Table 1: The results of spin operators.

The Hamiltonian in eq. (5.1) can be rewritten in terms of the raising and lowering operators

H_XXX = −J

N

X

j=1

(σ^x_jσ_j+1^x + σ^y_jσ^y_j+1+ σ_j^zσ^z_j+1) − B

N

X

j=1

σ_j^z

= −J

N

X

j=1

1

2(σ_j⁺σ⁻_j+1+ σ_j⁻σ_j+1⁺ ) + σ^z_jσ_j+1^z

− B

N

X

j=1

σ^z_j. (5.2)

The total z-component commutes with the Hamiltonian [P

iσ_i^z, H] = 0. The con- sequences of this are that the energy of the system is directly related to the total spin component, and that the contribution of the external magnetic field to the energy is just an additive constant. From now on, the external magnetic field will be omitted in the Ansatz.

Table 1 is used to calculate the ground state energy by applying the Hamiltonian to

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a spin chain in its ground state |ψi = |0i, H|ψi = E₀|ψi, resulting in

H|0i = −J

N

X

j=1

1

2(σ_j⁺σ_j+1⁻ + σ⁻_j σ_j+1⁺ ) + σ_j^zσ_j+1^z

|0i

= −J (0 + 0 +N

4)|0i = −J N 4 |0i E0 = −J N

4 . (5.3)

A chain in its ground state except for a single input qubit is said to be in the single excitation space. It refers to a chain with no more than one flipped spin, which is represented here again as |ji.

Ultimately, we want to come up with an expression for the Eigenvectors of this Hamil- tonian, which correspond to possible states of the spin chain. We guess an approximate form of these Eigenvectors using a basis of |ji to set up the Ansatz. This approximate form is

|Ψi =

N

X

j=1

a(j)|ji, (5.4)

where a(j) is a complex coefficient that varies with |ji. Plugging this into the Hamiltonian H|Ψi = E|Ψi, we get that

H|Ψi = −J

N

X

j=1 N

X

k=1

1

2(σ_k⁺σ_k+1⁻ + σ_k⁻σ_k+1⁺ ) + σ^z_kσ_k+1^z

a(j)|ji

= −J

N

X

j=1

a(j) 1

2|j − 1i +1

2|j + 1i + 1

4(N − 2) − 21 4

|ji

=

N

X

j=1

− J

2a(j + 1)|ji −J

2a(j − 1)|ji + a(j)

− N J 4 + J

|ji

,

(5.5)

which is just equal to the energy multiplied by the same Eigenvalues

E

N

X

j=1

a(j)|ji =

N

X

j=1

− J

2a(j + 1)|ji − J

2a(j − 1)|ji + a(j)

E₀+ J

|ji

E a(j) = −J

2a(j + 1) − J

2a(j − 1) + (E₀+ J )a(j). (5.6)

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Eq. (5.6) is a recursion relation for the Eigenvector coefficients. We impose a periodic boundary condition, a(g + N ) = a(g) to guess this recursion relation. Although this boundary condition better fits a spin ring, the boundary condition of a spin chain only creates a slight difference in the calculation with similar end results, and it is easier to follow from a ring boundary condition how those results are obtained. We make an Ansatz a(g) = Ae^imj, and using the boundary condition we determine that m = ^2π_N(k − 1), where k = 1, . . . , N . Plugging this in and simplifying, we get

EAe^imj = (E₀+ J )Ae^imj− AJ

2e^imje^im− AJ

2e^imje^−im (5.7)

⇔ E = (E₀+ J ) − AJ 2

e^im+ e^−im

(5.8)

⇔ E = E₀+ J − J cos(m) = E₀+ J [1 − cos ^2π_N(k − 1)] (5.9)

The allowed energies of this spin chain is given by E_n = E₀+ (1 − cos^2π_N(k − 1)), and the allowed Eigenvectors of the system are

|Ψ_ki = 1

√N

N

X

j=1

e^imj|ji, (5.10)

where A = ^√¹

N is a normalization constant. Note that these Eigenvectors e^imj are similar to a wave function describing a free particle. Excitations in the single excitation space are considered spin waves propagating through the spin chain. The information the chain is communicating about the qubit is a group of Gaussian wave packets, whose size and speed depends on the input qubit and size of the chain. This analogy to spin waves will prove useful in discussion of the spin chain dynamics.

The contribution of the magnetic field is just an additional term in the previous calculations. Repeating the same calculations,the ground state is

H_B|0i =

− J N

4 −BN 2

|0i = −(J + 2B)N

4|0i = E₀^B. (5.11) If we then do the same for the single single excitation space, we get one additional term in eq. (5.6),

−J

2a(j + 1) − J

2a(j − 1) + (E₀^B+ J + B)a(j) = E a(j). (5.12)

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The same exact Eigenvectors match this relation, and the Eigenvalues (allowed energies) are

E_k = E₀^B+ B + J 1 − cos^2π_N(k − 1). (5.13)

The Eigenvectors were found with periodic boundary conditions, which best fits a spin chain that formed a ring. For a spin chain that goes from point to point, a boundary condition that returns onto-itself is required. For a chain of length N = 10, then a(11) = a(10), and a(12) = a(9). This relation is written as a(N + i) = a(N + 1 − i). A cosine with period 2N and centered at N/2 − 1/2 matches this boundary condition

a(j) = A cos _2N^2π(j −¹₂)(k − 1) = A cos_N^π(j − ¹₂)(k − 1), (5.14)

where k goes from 1, . . . , N . Plugging this into the recursion relation of eq. (5.6), (after setting E0 = 0) and setting p = _N^π(k − 1) we get that

EA cos p(j − ¹₂) = JA cos p(j − ¹₂) − AJ 2

cos p(j − ¹₂ + 1) + cos p(j − ¹₂ − 1)

= J cos p(j −¹₂) − J 2

cos p(j −¹₂) cos(p) − sin p(j − ¹₂) sin(p) + cos p(j − ¹₂) cos(p) + sin p(j − ¹₂) sin(p)

= J cos p(j −¹₂) − J cos p(j − ¹₂) cos(p)

⇐⇒ E_k = J − J cos _N^π(k − 1)

(5.15)

Adding the magnetic field contribution as before, we find that the energies in the spin chain are

E_k = B + J1 − cos _N^π(k − 1). (5.16) Note that in eq. (5.16) the ground state energy is omitted. The Eigenvectors are expressed as

|ϕ_ki = a_k

N

X

j=1

cos _N^π(j − 1/2)(k − 1)|ji. (5.17)

The a_k is a normalization constant. It has values a_k = 1/√

N if k = 0, and a_k =p2/N when k 6= 0.

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5.2 Nearest Neighbor Fidelity

The total Fidelity of the spin chain is computed directly from the transition probability amplitude P_r of the receiving site. We take r to be the last spin of a chain of length N . Using the exact diagonalization of the Hamiltonian, we may write it in the form H =P

kE_k|ϕ_kihϕ_k|, where E_k are the Eigenvalues. The time evolution of an initial state is written as

|Ψ(t)i =X

k

hϕ_k|Ψ(0)ie^−iE^k^t/~|ϕ_ki (5.18) The inner product is taken to project our initial state onto the Eigenbasis of the Hamilto- nian. The transition probability amplitude eq. (4.7) can also be written in the Eigenbasis of the Hamiltonian,

P_r(t) =X

k

hr|ϕ_kihϕ_k|sie^−iE^k^t/~. (5.19)

Note that the Eigenvectors |ϕ_ki are written as a summation over all |ji states. The sender and receiver sites are the first and last sites of the spin chain, and they are orthogonal to all other |ji states. So, the only terms that survive the inner product projecting them onto the Eigenbasis are the first and last ones. The inner product hr|ϕ_ki explicitly written out is

hr|ϕ_ki = a_k

N

X

j=1

cos _N^π(j − ¹₂)(k − 1)hN|ji hr|ϕ_ki = a_kcos _N^π(N − ¹₂)(k − 1)

(5.20)

By similar logic we get that hϕ_k|si = a_kcos _2N^π (k − 1). Eq. (5.19) may be rewritten as

P_r(t) =X

k

a²_kcos _N^π(N −¹₂)(k − 1) cos _2N^π (k − 1)e^−iE^k^t/~. (5.21)

This analytical diagonalization of our Hamiltonian has simplified the problem of finding the transition probability from an exponentially increasing Hilbert space to a summation over cosines. From here, the Fidelity-squared is calculated according to eq. (4.20). The one part that requires extra attention is the arg(P_r) in the expression. In order to maximize the total average fidelity, we need to ensure that arg[cos(P_r)] is a multiple of 2π when the transition probability is at a maximum. Since the only part of our equation

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that contributes to the argument is e^iE^k^t/~, the magnetic field is used to ensure this maximum.

0.5 1.0 1.5 2.0 B

-1.0 -0.5 0.5 1.0 Cos(Arg(P))

Figure 2: For spin chain N = 3, values of cos arg(P_r(t₀)) for varying external magnetic fields.

This is done by first finding the maximum value of our transition probability amplitude within a predefined range of t, then maximizing the argument of cos(P_r) at that point.

As we see in figure 2, there are a series of optimal values of B for the argument in eq. (4.20). Changing the magnetic field does not affect the transition probability because the external field is just a constant added to all energies eq. (5.16). Exponentiating the energies in eq. (5.21) to describe time evolution makes the magnetic field an overall phase for all the Eigenvalues.

P_r(t) =X

k

hr|ϕ_kihϕ_k|sie^−iE^k^t/~

=X

k

hr|ϕ_kihϕ_k|sie−i[B+J[1−cos(π

N(k−1))]]t/~

=X

k

hr|ϕ_kihϕ_k|sie^{−iJ[1−cos(}^N^π^(k−1))]t/~e^−iBt/~

P_r(t) = e^−iBt/~X

k

hr|ϕ_kihϕ_k|sie^{−iJ[1−cos(}^N^π^(k−1))]t/~. (5.22)

The main contribution of the magnetic field to the total average fidelity comes from this argument. The nature of this contribution is akin to a resonant frequency for sending quantum information through the spin chain. As shown in figure 2, there is a related series of magnetic field strengths that best aid in the transfer of quantum information.

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0 10 20 30 40 0.0

0.2 0.4 0.6 0.8 1.0

t

P(t),F(t)

Spin chain of size 2

0 20 40 60 80 100

0.0 0.2 0.4 0.6 0.8 1.0

t

P(t),F(t)

Spin chain of size 3

Figure 3: Periodic transition probabilities (dashed, blue) and total average fidelity (solid, orange) of spin chains plotted over time. For a spin chain of size N the fidelity was maximized within a range of t ∈ [0, 5πN ln(N )]. For the chain N = 2 there is periodic perfect state transfer.

For the cases N = 2, 3 the transition probability and total average fidelity have a very repetitive structure, as shown in figure 3. However, for spin chains of size N > 3, there is no symmetry or periodicity, as shown in figure 4. If the total average fidelity is equal to one at a time t₀, the state of the receiving site is exactly that of the input qubit and out spin chain is in the state |Ψ(t₀)i = cos^θ₂|0i + e^iϕsin^θ₂|ri. This is called perfect state transfer. We see that in the case where N = 2, 4 there is perfect state transfer.

The results in Figures 3 and 4 match the intuition that comes from thinking of Eigen- vectors as spin waves. More constructive interference of the spin waves at the receiving end of the spin chain corresponds to more information about the input state. However, as the spin chains become larger there is a wider range of spin waves propagating through

Quantum Speed Limits of Spin Chain Dynamics: Applying Quantum Speed Limits to Spin Chains used as Quantum Channels