Quantum State Transfer with Spin Chains

TL;DR

This paper explores methods for transferring quantum states across spin chains with permanent couplings, aiming to improve quantum communication efficiency.

Contribution

It provides a comprehensive analysis of quantum state transfer techniques in spin chains, highlighting new protocols for reliable transfer.

Findings

Identified optimal conditions for high-fidelity transfer

Developed a new protocol with improved robustness

Demonstrated potential for scalable quantum networks

Abstract

The thesis covers various aspects of quantum state transfer in permanently coupled spin systems.

Figures

Tables

Table 1. Table 1: The total time t 𝑡 t and the number of measurements M 𝑀 M needed to achieve a probability of success of 99 99 99 % for different fluctuation strengths Δ Δ \Delta (uncorrelated case). Given is the statistical mean and the standard deviation. The length of the chain is N = 20 𝑁 20 N=20 and the number of random samples is 10 . 10 10. For strong fluctuations Δ = 0.1 Δ 0.1 \Delta=0.1 , we also found particular samples where the success probability could not be achieved within the time range searched by the algorithm.

	$\Delta =0$	$\Delta =0.01$	$\Delta =0.03$	$\Delta =0.05$	$\Delta =0.1$
$t$$\left[\frac{1}{J}\right]$	$377$	$524\pm 27$	$694\pm 32$	$775\pm 40$	$1106\pm 248$
$M$	$28$	$43\pm 3$	$58\pm 3$	$65\pm 4$	$110\pm 25$

Table 2. Table 2: The total time t 𝑡 t and the number of measurements M 𝑀 M needed to achieve a probability of success of 99 99 99 % for different correlations c 𝑐 c between the couplings [see Eq. ( 98 ) and Eq. ( 99 )]. Given is the statistical mean and the standard deviation for a fluctuation strength of Δ = 0.05 Δ 0.05 \Delta=0.05 . The length of the chain is N = 20 𝑁 20 N=20 and the number of random samples is 20 . 20 20.

	$c=0$	$c=0.1$	$c=0.3$	$c=0.7$	$c=0.9$	$c=1$
$t$$\left[\frac{1}{J}\right]$	$666\pm 20$	$725\pm 32$	$755\pm 41$	$797\pm 35$	$882\pm 83$	$714\pm 41$
$M$	$256\pm 2$	$62\pm 3$	$65\pm 4$	$67\pm 4$	$77\pm 7$	$60\pm 4$

Equations

$$\alpha |0⟩+\beta |1⟩,$$

$$|G⟩.$$

$$\rho \otimes \text{mbox}{Tr}_1\left\{|G⟩⟨G|\right\}.$$

$$\exp \left\{-iHt\right\}\rho \otimes \text{mbox}{Tr}_1\left\{|G⟩⟨G|\right\}\exp \left\{iHt\right\}.$$

$$\sigma \equiv \text{mbox}{Tr}_{1,\dots ,N-1}\left[\exp \left\{-iHt\right\}\rho \otimes \text{mbox}{Tr}_1\left\{|G⟩⟨G|\right\}\exp \left\{iHt\right\}\right].$$

$$F(\rho ,\sigma )\equiv {\left(\text{mbox}Tr\sqrt{{\rho }^{1/2}\sigma {\rho }^{1/2}}\right)}^2.$$

$$F_0\equiv \text{mbox}{min}_{\rho }F(\rho ,\tau (\rho )).$$

$$F_0=\text{mbox}{min}_{\psi }⟨\psi |\tau (\psi )|\psi ⟩.$$

$$\mathcal{H}=\underset{\ell =0}{\overset{N}{⨁}}{\mathcal{H}}_{\ell },$$

$$|G⟩=|0⟩.$$

$$|0⟩$$

$$\alpha |0⟩+\beta |1⟩$$

$$\alpha |0⟩+\beta \exp \left\{-iHt\right\}|1⟩$$

$$\tau (\psi )=(1-p(t))|0⟩⟨0|+p(t)|\psi ⟩⟨\psi |,$$

$$F_0$$

$$=p(t)+(1-p(t))\text{mbox}{min}_{\psi }{\left|⟨0|\psi ⟩\right|}^2=p(t).$$

$$p(t)={\left|⟨\mathbf{N}|\exp \left\{-iHt\right\}|1⟩\right|}^2.$$

$$\bar{F}=\frac{\sqrt{p(t)}}{3}+\frac{p(t)}{6}+\frac{1}{2}.$$

$$f_{n,m}(t)$$

$$=\sum _k{e}^{-i{E}_{k}t}⟨\mathbf{n}|E_k⟩⟨E_k|\mathbf{m}⟩,$$

$$H=-\frac{J}{2}\sum _{n=1}^{N-1}\left(X_n{X}_{n+1}+Y_n{Y}_{n+1}+Z_n{Z}_{n+1}\right)-B\sum _{n=1}^N{Z}_n+c,$$

$$c=\frac{J(N-1)}{2}+NB$$

$$\left(\begin{array}[]{cccccc}1&-1\\ -1&2&-1\\ &-1&2&\ddots\\ &&\ddots&\ddots&-1\\ &&&-1&2&-1\\ &&&&-1&1\end{array}\right).$$

$$|E_k⟩=\sqrt{\frac{1+{\delta }_{k0}}{N}}\sum _{n=1}^N\cos \left[\frac{\pi k}{2N}(2n-1)\right]|\mathbf{n}⟩(k=0,\dots ,N-1),$$

$$E_k=2B+2J\left[1-\cos \frac{\pi k}{N}\right].$$

$$\boxed{p(t)=N^{-2}{\left|1+\sum _{k=1}^{N-1}\exp \left\{-2iJt(1-\cos \frac{\pi k}{N})\right\}(-1{)}^k\left(1+\cos \frac{\pi k}{N}\right)\right|}^2.}$$

$$t_{\text{mbox}peak}\approx \frac{N}{2J}$$

$$|f_{N,1}(t){|}^2\approx |2J_N(\frac{2t}{J}){|}^2\approx |{\left(\frac{16}{N}\right)}^{1/3}\text{mbox}ai[{\left(\frac{2}{N}\right)}^{1/3}(N-\frac{2t}{J})]{|}^2,$$

$$p(t_{\text{mbox}peak})=|f_{N,1}(\frac{N}{2J}){|}^2\approx 1.82N^{-2/3}.$$

$$p_M(T)\equiv \underset{0<t<T}{\max }p(t).$$

$$\left\{\cos \frac{k\pi }{N}(k=0,1,\dots ,,\frac{N-1}{2})\right\}$$

Full text

\typearea

[20mm]12

Quantum State Transfer with Spin Chains

Daniel Klaus Burgarth

A thesis submitted to the University of London

for the degree of Doctor of Philosophy

Department of Physics and Astronomy

University College London

December 2006

Declaration

I, Daniel Klaus Burgarth, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis.

Abstract

In the last few decades the idea came up that by making use of the superposition principle from Quantum Mechanics, one can process information in a new and much faster way. Hence a new field of information technology, QIT (Quantum Information Technology), has emerged. From a physics point of view it is important to find ways of implementing these new methods in real systems. One of the most basic tasks required for QIT is the ability to connect different components of a Quantum Computer by quantum wires that obey the superposition principle. Since superpositions can be very sensitive to noise this turns out to be already quite difficult. Recently, it was suggested to use chains of permanently coupled spin-1/2 particles (quantum chains) for this purpose. They have the advantage that no external control along the wire is required during the transport of information, which makes it possible to isolate the wire from sources of noise. The purpose of this thesis is to develop and investigate advanced schemes for using quantum chains as wires. We first give an introduction to basic quantum state transfer and review existing advanced schemes by other authors. We then introduce two new methods which were created as a part of this thesis. First, we show how the fidelity of transfer can be made perfect by performing measurements at the receiving end of the chain. Then we introduce a scheme which is based on performing unitary operations at the end of the chain. We generalise both methods and discuss them from the more fundamental point of view of mixing properties of a quantum channel. Finally, we study the effects of a non-Markovian environment on quantum state transfer.

Acknowledgements

Most of all, I would like to thank my supervisor Sougato Bose for much inspiration and advice. I am very grateful for many inspiring and fruitful discussions and collaborations with Vittorio Giovannetti, and with Floor Paauw, Christoph Bruder, Jason Twamley, Andreas Buchleitner and Vladimir Korepin. Furthermore I would like to thank all my teachers and those who have guided and motivated me along my journey through physics, including Heinz-Peter Breuer, Francesco Petruccione, Lewis Ryder, John Strange, Werner Riegler, Carsten Schuldt and Rolf Bussmann. I acknowledge financial support by the UK Engineering and Physical Sciences Research Council through the grant GR/S62796/01. Finally I would like to thank my parents for their loving support.

Notation

X,Y,Z Pauli matrices

$X_{n},Y_{n},Z_{n}$ Pauli matrices acting on the Hilbert-space of qubit $n$

$|0\rangle,|1\rangle$ Single qubit state in the canonical basis

$|\boldsymbol{0}\rangle$ Quantum chain in the product state $|0\rangle\otimes\cdots\otimes|0\rangle$

$|\boldsymbol{n}\rangle$ ”Single excitation” state $X_{n}|\boldsymbol{0}\rangle$

$\mbox{Tr}_{X}$ Partial trace over subsystem $X$

$||\ldots||$ Euclidean vector norm

$||\ldots||_{1}$ Trace norm

$||\ldots||_{2}$ Euclidean matrix norm

We also use the following graphical representation:

Contents

1. 1 Introduction

2. 1 Quantum Computation and Quantum Information

3. 2 Quantum state transfer along short distances

4. 3 Implementations and experiments

5. 4 Basic communication protocol

6. 4.1 Initialisation and end-gates

7. 4.2 Symmetries

8. 4.3 Transfer functions

9. 4.4 Heisenberg Hamiltonian

10. 4.5 Dynamic and Dispersion

11. 4.6 How high should $p(t)$ be?

12. 5 Advanced communication protocols

13. 5.1 Engineered Hamiltonians

14. 5.2 Weakly coupled sender and receiver

15. 5.3 Encoding

16. 5.4 Time-dependent control

17. 6 Motivation and outline of this work

18. 2 Dual Rail encoding

19. 7 Introduction

20. 8 Scheme for conclusive transfer

21. 9 Arbitrarily perfect state transfer

22. 10 Estimation of the time-scale the transfer

23. 11 Decoherence and imperfections

24. 12 Disordered chains

25. 13 Conclusive transfer in the presence of disorder

26. 14 Arbitrarily perfect transfer in the presence of disorder

27. 15 Tomography

28. 16 Numerical Examples

29. 17 Coupled chains

30. 18 Conclusion

31. 3 Multi Rail encoding

32. 19 Introduction

33. 20 The model

34. 21 Efficient encoding

35. 22 Perfect transfer

36. 23 Convergence theorem

37. 24 Quantum chains with nearest-neighbour interactions

38. 25 Comparison with Dual Rail

39. 26 Conclusion

40. 4 Ergodicity and mixing

41. 27 Introduction

42. 28 Topological background

43. 29 Generalised Lyapunov Theorem

44. 29.1 Topological spaces

45. 29.2 Metric spaces

46. 30 Quantum Channels

47. 30.1 Mixing criteria for Quantum Channels

48. 30.2 Beyond the density matrix operator space: spectral properties

49. 30.3 Ergodic channels with pure fixed points

50. 31 Conclusion

51. 5 Read and write access by local control

52. 32 Introduction

53. 33 Protocol

54. 34 Decomposition equations

55. 35 Coding transformation

56. 36 Fidelities for reading and writing

57. 37 Application to spin chain communication

58. 38 Conclusion

59. 6 A valve for probability amplitude

60. 39 Introduction

61. 40 Arbitrarily Perfect State Transfer

62. 41 Practical Considerations

63. 42 Conclusion

64. 7 External noise

65. 43 Introduction

66. 44 Model

67. 45 Results

68. 46 Conclusion

69. 8 Conclusion and outlook

Chapter 1 Introduction

The Hilbert space that contains the states of quantum mechanical objects is huge, scaling exponentially with the number of particles described. In 1982, Richard Feynman suggested to make use of this as a resource for simulating quantum mechanics in a quantum computer, i.e. a device where the physical interaction could be “programmed” to yield a specific Hamiltonian. This has led to the new fields of Quantum Computation and Quantum Information. A quantum computer can solve questions one could never imagine to solve using an ordinary computer. For example, it can factorise large numbers into primes efficiently, a task of greatest importance for cryptography. It may thus be a surprise that more than twenty years after the initial ideas, these devices still haven’t been built or only in ridiculously small size. The largest quantum computer so far can only solve problems that any child could solve within seconds. A closer look reveals that the main problem in the realisation of quantum computers is the “programming”, i.e. the design of a specific (time-dependent) Hamiltonian, usually described as a set of discrete unitary gates. This turns out to be extremely difficult because we need to connect microscopic objects (those behaving quantum mechanically) with macroscopic devices that control the microscopic behaviour. Even if one manages to find a link between the micro- and the macroscopic world, such as laser pulses and electric or magnetic fields, then the connection introduces not only control but also noise (dissipation and decoherence) to the microscopic system, and its quantum behaviour is diminished.

The vision of this thesis is to develop theoretical methods narrowing the gap between what is imagined theoretically and what can be done experimentally. As a method we consider chains (or more general graphs) of permanently coupled quantum systems. This idea has been originally put forward by S. Bose for the specific task of quantum communication [1]. Due to the permanent coupling, these devices can in principle be built in such a way that they don’t require external control to perform their tasks, just like a mechanical clockwork. This also overcomes the problem of decoherence as they can be separated from any source of noise. Unfortunately, most schemes that have been developed so far still require external control, though much less than an “ordinary” quantum computer. Furthermore, internal dispersion in these devices is leading to a decrease of their fidelity. A third problem is, that for building these devices the permanent couplings still need to be realised, although only once, and experimental constraints such as resolution and errors need to be considered. We are thus left with the following questions: which is the best way to perform quantum state transfer using a permanently coupled graph? How much control do we need, and how difficult will it be to implement the couplings? How do errors and noise affect the scheme? All these points are highly related and it cannot be expected to find an absolute, i.e. system independent answer. The purpose of this research is to develop advanced schemes for the transfer of quantum information, to improve and generalise existing ideas, to relate them to each other and to investigate their stability and efficiency.

1 Quantum Computation and Quantum Information

In this Section we review some of the basic concepts of Quantum Computation. We will be very brief and only focus on those aspects that we require later on in the thesis. A more detailed introduction can be found in [2].

In information science, an algorithm is a list of instructions that a computer performs on a given input to achieve a specific task. For instance, a factoring algorithm has an arbitrary integer as its input, and gives its prime factors as an output. A quantum factoring algorithm can be thought of in a similar way, i.e. it has an integer as input, and its prime factors as an output. In-between however it encodes information in a quantum mechanical system. Due to the superposition principle, the information of a quantum system cannot be represented as bits. The valid generalisation of the bit to the quantum case is called qubit. The possible states of a qubit are written as

[TABLE]

where $\alpha,\beta$ are normalised complex coefficients, and $|0\rangle$ and $|1\rangle$ are vectors of a two-dimensional complex vector space. Peter W. Shor has shown in a famous paper [3] that the detour of representing the intermediate part of a factoring algorithm in a quantum system (as well as using quantum gates, see below) can be very beneficial: it runs much faster. This is important, because many cryptographic methods rely on factoring algorithms being slow. Shor’s algorithm is definitely not the only reason why it would be very nice to have a quantum computer, i.e. a machine that represents information in a quantum way and can perform instructions on it, and many more details can be found in the textbook mentioned above.

Algorithms on a computer can be represented as list of logical operations on bits. Likewise, a (standard) quantum algorithm can be represented as a list of quantum logical operations, or quantum gates, acting on qubits. The most general quantum algorithm is given by an arbitrary unitary operator. A universal set of gates is a set such that any quantum algorithm (i.e. unitary operator) can be decomposed into a sequence of gates belonging to this set. In the standard model of quantum computation, one assumes that such a set is available on the machine [4]. Also the ability to perform measurements is assumed. We refer to this as the full control case.

From a information theoretic point of view, qubits are not only useful objects to perform algorithms with, but also very interesting from a fundamental point of view. To give a (too simple) analogy consider the following. If you read the word ”chocolate”, you can associate a positive/negative or neutral feeling of whether you would like to eat some chocolate now. However, what was the state of your mind concerning chocolate before you read the word? **Unless you were already craving for chocolate beforehand, or you have just eaten a lot, your mind was probably undecided. Moreover, it would have been very difficult - if not impossible - to describe to someone in plain language which opinion you had about the chocolate before you read the word.

In a similar manner, the quantum information contained in a single arbitrary and unknown qubit cannot be described by classical information. When it is measured, it behaves like a normal bit in the sense that the outcome is only [math] or $1,$ but when it is not measured, it behaves in some way as if it was undecided between [math] and 1. Of course one has to be very careful with these analogies. But for the purpose of this thesis it is important to stress that quantum information cannot be transported by any classical methods [5]. This is why it is so important and also so difficult to develop new wires, dubbed quantum wires, that are capable of doing this.

2 Quantum state transfer along short distances

In theory, additional devices for the transfer of unknown quantum states are not required for building a quantum computer, unless it is being used for typical quantum communication purposes, such as secret key distribution [4]. This is because the universal set of gates on the quantum computer can be used to transfer quantum states by applying sequences of two-qubit swap gates (Fig. 1).

However in practice it is crucial to minimise the required number of quantum gates, as each gate typically introduces errors. In this light it appears costly to perform $N-1$ swap gates between nearest neighbours to just move a qubit state over a distance of $N$ sites. For example, Shor’s algorithm on $N$ qubits can be implemented by only $\log N$ quantum gating operations [6] if long distant qubit gates are available. These long distant gates could consist of local gates followed by a quantum state transfer. If however the quantum state transfer is implemented as a sequence of local gates, then the number of operations blows up to the order of $N$ gates. The quantum state transfer can even be thought of as the source of the power of quantum computation, as any quantum circuit with $\log N$ gates and local gates only can be efficiently simulated on a classical computer [7, 8].

A second reason to consider devices for quantum state transfer is related to scalability. While small quantum computers have already been built [9], it is very difficult to build large arrays of fully controllable qubits. A black box that transports unknown quantum states could be used to build larger quantum computers out of small components by connecting them. Likewise, quantum state transfer can be used to connect different components of a quantum computer, such as the processor and the memory (see also Fig. 2). On larger distances, flying qubits such as photons, ballistic electrons and guided atoms/ions are considered for this purpose [10, 11]. However, converting back and forth between stationary qubits and mobile carriers of quantum information and interfacing between different physical implementations of qubits is very difficult and worthwhile only for short communication distances. This is the typical situation one has to face in solid state systems, where quantum information is usually contained in the states of fixed objects such as quantum dots or Josephson junctions. In this case permanently coupled quantum chains have recently been proposed as prototypes of reliable quantum communication lines [1, 12]. A quantum chain (also referred to as spin chain) is a one-dimensional array of qubits which are coupled by some Hamiltonian (cf. Fig. 3). These couplings can transfer states without external classical control. In many cases, such permanent couplings are easy to build in solid state devices (in fact a lot of effort usually goes into suppressing them). The qubits can be of the same type as the other qubits in the device, so no interfacing is required.

Another related motivation to consider quantum chains is that they can simplify the layout of quantum devices on wafers. A typical chip can contain millions of qubits, and the fabrication of many qubits is in principle no more difficult than the fabrication of a single one. In the last couple of years, remarkable progress was made in experiments with quantum dots [13, 14] and super-conducting qubits [15, 16]. It should however be emphasised that for initialisation, control and readout, those qubits have to be connected to the macroscopic world (see Fig. 2). For example, in a typical flux qubit gate, microwave pulses are applied onto specific qubits of the sample. This requires many (classical) wires on the chip, which is thus a compound of quantum and classical components. The macroscopic size of the classical control is likely to be the bottleneck of the scalability as a whole. In this situation, quantum chains are useful in order to keep some distance between the controlled quantum parts. A possible layout for such a quantum computer is shown in Fig. 4. It is built out of blocks of qubits, some of which are dedicated to communication and therefore connected to another block through a quantum chain. Within each block, arbitrary unitary operations can be performed in a fast and reliable way (they may be decomposed into single and two-qubit operations). Such blocks do not currently exist, but they are the focus of much work in solid state quantum computer architecture. The distance between the blocks is determined by the length of the quantum chains between them. It should be large enough to allow for classical control wiring of each block, but short enough so that the time-scale of the quantum chain communication is well below the time-scale of decoherence in the system.

Finally, an important reason to study quantum state transfer in quantum chains stems from a more fundamental point of view. Such systems in principle allow tests of Bell-inequalities and non-locality in solid-state experiments well before the realisation of a quantum computer. Although quantum transport is quite an established field, the quantum information point of view offers many new perspectives. Here, one looks at the transport of information rather than excitations, and at entanglement [17, 18, 19, 20] rather than correlation functions. It has recently been shown that this sheds new light on well-known physical phenomena such as quantum phase transitions [21, 22, 23, 24], quantum chaos [25, 26, 27, 28] and localisation [29, 30]. Furthermore, quantum information takes on a more active attitude. The correlations of the system are not just calculated, but one also looks at how they may be changed.

3 Implementations and experiments

As we have seen above, the main advantage of state transfer with quantum chains is that the qubits can be of the same type as those used for the quantum computation. Therefore, most systems that are thought of as possible realisations of a quantum computer can also be used to build quantum chains. Of course there has to be some coupling between the qubits. This is typically easy to achieve in solid state systems, such as Josephson junctions with charge qubits [31, 32], flux qubits [33, 34] (see also Fig. 5) or quantum dots dots using the electrons [35, 36] or excitons [37, 38]. Other systems where quantum chain Hamiltonians can at least be simulated are NMR qubits [39, 40, 41] and optical lattices [42]. Such a simulation is particularly useful in the latter case, where local control is extremely difficult. Finally, qubits in cavities [43, 44] and coupled arrays of cavities were considered [45, 46].

For the more fundamental questions, such as studies of entanglement transfer, non-locality and coherent transport, the quantum chains could also be realised by systems which are not typically thought of as qubits, but which are natural spin chains. These can be molecular systems [47] or quasi-1D solid state materials [48, 49].

4 Basic communication protocol

We now review the most basic transport protocol for quantum state transfer, initially suggested in [1]. For the sake of simplicity, we concentrate on the linear chain setting, though more general graphs of qubits can be considered in the same way. The protocol consists of the following steps:

Initialise the quantum chain in the ground state

[TABLE] 2. 2.

Put an arbitrary and unknown qubit with (possibly mixed) state $\rho$ at the sending end of the chain

[TABLE] 3. 3.

Let the system evolve under its Hamiltonian $H$ for a time $t$

[TABLE] 4. 4.

Pick up the quantum state at the end of the chain

[TABLE]

Some practical aspects how to realise these steps are discussed in the next section. For the moment, we will concentrate on the quality of quantum state transfer given that the above steps can be performed. From a quantum information perspective, the above equations describe a quantum channel [5] $\tau$ that maps input states $\rho$ at one end of the chain to output states $\tau(\rho)=\sigma$ on the other end. A very simple measure of the quality of such a quantum channel is the fidelity [50, 51, 2]**

[TABLE]

More advanced measures of the quality of transfer will be discussed in Chapter 3. Note also that some authors define the fidelity without taking the square of the trace. It is a real-valued, symmetric function with range between [math] and $1,$ assuming unity if and only if $\rho=\sigma.$ Since the transported state that is an unknown result of some quantum computation, we are interested in the minimal fidelity

[TABLE]

We remark that some authors also assume an equal distribution of input states and compute the average fidelity [1]. Using the strong concavity of the fidelity [2] and the linearity of $\tau$ we find that the minimum must be assumed on pure input states,

[TABLE]

In the present context, $F_{0}=F_{0}(H,t)$ is a function of of the Hamiltonian $H$ of the quantum chain (through the specific role of the ground state in the protocol and through the time evolution), and of the time interval $t$ that the system is evolving in the third step of the protocol.

4.1 Initialisation and end-gates

There are two strong assumptions in the protocol from the last section. The first one is that the chain can be initialised in the ground state $|G\rangle.$ How can that be achieved if there is no local control along the chain? The answer appears to be quite easy: one just applies a strong global magnetic field and strong cooling (such as laser cooling or dilution refrigeration) and lets the system reach its ground state by relaxation. The cooling needs to be done for the remaining parts of the quantum computer anyway, so no extra devices are required. However there is a problem with the time-scale of the relaxation. If the system is brought to the ground state by cooling, it must be coupled to some environment. But during the quantum computation, one clearly does not want such an environment. This is usually solved by having the time-scale of the computation much smaller (say microseconds) than the time-scale of the cooling (say seconds or minutes). But if the quantum chain should be used multiple times during one computation, then how is it reset between each usage? This is important to avoid memory effects [52], and there are two solutions to this problem. Either the protocol is such that at the end the chain is automatically in the ground state. Such a protocol usually corresponds to perfect state transfer. The other way is to use the control at the ends of the chain to bring it back to the ground state. A simple cooling protocol is given by the following: one measures the state of the last qubit of the chain. If it is in $|0\rangle,$ then one just lets the chain evolve again and repeats. If however it is found to be in $|1\rangle,$ one applies the Pauli operator $X$ to flip it before evolving and repeating. It will become clear later on in the thesis that such a protocol typically converges exponentially fast to the ground state of the chain.

The second assumption in the last section is that the sender and receiver are capable of swapping in and out the state much quicker than the time-scale of the interaction of the chain. Alternatively, it is assumed that they can switch on and off the interaction between the chain and their memory in such time-scale. It has recently been shown [33] that this is not a fundamental problem, and that finite switching times can even slightly improve the fidelity if they are carefully included in the protocol. But this requires to solve the full time-dependent Schrödinger equation, and introduces further parameters to the model (i.e. the raise and fall time of the couplings). For the sake of simplicity, we will therefore assume that the end gates are much faster then the time evolution of the chain (see also Section 41).

4.2 Symmetries

The dimensionality of the Hilbert space $\mathcal{H}$ of a quantum chain of $N$ qubits is $2^{N}.$ This makes it quite hopeless in general to determine the minimal fidelity Eq. (8) for long quantum chains. Most investigations on quantum state transfer with quantum chains up to date are therefore concentrating on Hamiltonians with additional symmetries. With few exceptions [34, 21, 22, 53] Hamiltonians that conserve the number of excitations are considered. In this case the Hilbert space is a direct sum of subspaces invariant under the time evolution,

[TABLE]

with $\mbox{dim}\mathcal{H}_{\ell}=\binom{N}{\ell},$ and where $\ell$ is the number of excitations. These Hamiltonians are much easier to handle both analytically and numerically, and it is also easier to get an intuition of the dynamics. Furthermore, they occur quite naturally as a coupling between qubits in the relevant systems. We stress though that there is no fundamental reason to restrict quantum chain communication to this case.

4.3 Transfer functions

The space $\mathcal{H}_{0}$ only contains the state $|\boldsymbol{0}\rangle$ which is thus always an eigenstate of $H.$ We will assume here that it is also the ground state,

[TABLE]

This can be achieved by applying a strong global magnetic field (or equivalent) to the system. The space $\mathcal{H}_{1}$ is spanned by the vectors $\left\{|\boldsymbol{k}\rangle,k=1,\ldots,N\right\}$ having exactly one excitation. The above protocol becomes:

Initialise the quantum chain in the ground state

[TABLE] 2. 2.

Put an arbitrary and unknown qubit in the pure state $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ at the sending end of the chain

[TABLE] 3. 3.

Let the system evolve for a time $t$

[TABLE] 4. 4.

Pick up the quantum state at the end of the chain (see [1])

[TABLE]

with the minimal fidelity given by

[TABLE]

The function $p(t)$ is the transition probability from the state $|\boldsymbol{1}\rangle$ to $|\boldsymbol{N}\rangle$ given by

[TABLE]

We see that in the context of quantum state transfer, a single parameter suffices to characterise the properties of an excitation conserving chain. The averaged fidelity [1] is also easily computed as

[TABLE]

Even more complex measures of transfer such as the quantum capacity only depend on $p(t)$ [54]. It is also a physically intuitive quantity, namely a particular matrix element of the time evolution operator,

[TABLE]

where $|E_{k}\rangle$ and $E_{k}$ are the eigenstates and energy levels of the Hamiltonian in $\mathcal{H}_{1}.$

4.4 Heisenberg Hamiltonian

The Hamiltonian chosen in [1] is a Heisenberg Hamiltonian

[TABLE]

with a constant term

[TABLE]

added to set the ground state energy to $0.$ For $J>0$ it fulfils all the assumptions discussed above, namely its ground state is given by $|\boldsymbol{0}\rangle$ and it conserves the number of excitations in the chain. The Heisenberg interaction is very common and serves here as a typical and analytically solvable model for quantum state transfer.

In the first excitation subspace $\mathcal{H}_{1}$, the Heisenberg Hamiltonian Eq. (21) is expressed in the basis $\left\{|\boldsymbol{n}\rangle\right\}$ as

[TABLE]

A more general study of such tridiagonal matrices can be found in a series of articles on coherent dynamics [55, 56, 57, 58]. Some interesting analytically solvable models have also been identified [59, 56, 57] (we shall come back to that point later).

For the present case, the eigenstates of Eq (23) are [1]

[TABLE]

with the corresponding energies given by

[TABLE]

The parameter $B$ has no relevance for the fidelity but determines the stability of the ground state (the energy of the first excited state is given by $2B).$ The minimal fidelity for a Heisenberg chain is given by

[TABLE]

As an example, Fig 6 shows $p(t)$ for $N=50$.

4.5 Dynamic and Dispersion

Already in [1] has been realised that the fidelity for quantum state transfer along spin chains will in general not be perfect. The reason for the imperfect transfer is the dispersion [60] of the information along the chain. Initially the quantum information is localised at the sender, but as it travels through the chain it also spreads (see Fig. 7 and Fig. 8). This is not limited to the Heisenberg coupling considered here, but a very common quantum effect. Due to the dispersion, the probability amplitude peak that reaches Bob is typically small, and becomes even smaller as the chains get longer.

The fidelity given Eq. (26) is shown in Fig. 6. We can see that a wave of quantum information is travelling across the chain. It reaches the other end at a time of approximately

[TABLE]

As a rough estimate of the scaling of the fidelity with respect to the chain length around this peak we can use [1, 61] (see also Fig. 9)

[TABLE]

where $J_{N}(x)$ is a Bessel function of first kind and $ai(x)$ is the Airy function. The airy function $ai(x)$ has a maximum of $0.54$ at $x=-1.02.$ Hence we have

[TABLE]

It is however possible to find times where the fidelity of the chain is much higher. The reason for this is that the wave-packet is reflected at the ends of the chain and starts interfering with itself (Fig 6). As the time goes on, the probability distribution becomes more and more random. Sometimes high peaks at the receiving end occur. From a theoretical point of view, it is interesting to determine the maximal peak occurring, i.e.

[TABLE]

As we can see in Fig. 10 there is quite a potential to improve from the estimate Eq. (29).

We will now show a perhaps surprising connection of the function $p_{M}(T)$ to number theory. Some speculations on the dependence of the fidelity on the chain length being divisible by $3$ were already made in [1], but not rigorously studied. As it turns out, for chains with prime number length the maximum of the fidelity is actually converging to unity (see Fig. 10). To show this, we first prove the following

Lemma 1.1**.**

Let $N$ be an odd prime. Then the set

[TABLE]

*is linear independent over the rationals $\mathbb{Q}$. *

Proof.

Assume that

[TABLE]

with $\lambda_{k}\in\mathbb{Q}.$ It follows that

[TABLE]

and hence

[TABLE]

Changing indexes on the second sum we get

[TABLE]

and finally

[TABLE]

where

[TABLE]

Since $N$ is prime, the roots of unity in Eq. (36) are all primitive and therefore linearly independent over $\mathbb{Q}$ [62, Theorem 3.1, p. 313]. Hence $\lambda_{k}=0$ for all $k.$ ■

Theorem 1.1** (Half recurrence).**

Let $N$ be an odd prime. For a Heisenberg chain of length $N$ we have

[TABLE]

Proof.

The eigenfrequencies of the Hamiltonian in the first excitation sector $\mathcal{H}_{1}$ are given by

[TABLE]

Using Kronecker’s theorem [63] and Lemma 1.1, the equalities

[TABLE]

can be fulfilled arbitrarily well by choosing an appropriate $t.$ Since

[TABLE]

the equalities (42) are then also fulfilled arbitrarily well for $k=0,\ldots,N-1.$ This is known as as sufficient condition for perfect state transfer in mirror symmetric chains [64], where the eigenstates can be chosen such that they are alternately symmetric and antisymmetric. Roughly speaking, Eq. (42) introduces the correct phases (a sign change for the antisymmetric eigenstates) to move the state $|1\rangle$ to $|N\rangle$ and hence the theorem. ■

Remark 1.1*.*

The time-scale for finding high valued peaks is however exponential in the chain length [63]. Therefore the above theorem has little practical use. For non-prime chain lengths, the eigenfrequencies are not sufficiently independent to guarantee a perfect state transfer, with the algebraic dimensionality of the roots of unity for non-prime $N$ given by the Euler totient function $\phi(N)$ [62, Theorem 3.1, p. 313]. We also remark that due to its asymptotic character, the above result is not contradicting [65], where it was shown that chains longer than $N\geq 4$ never have perfect fidelity.

Having proved that there are many chains that can in principle perform arbitrarily well, it is important to find a cut-off time for the optimisation Eq. (30). Faster transfer than linear in $N$ using local Hamiltonians is impossible due to the Lieb-Robinson bound [66, 67], which is a ”speed limit” in non-relativistic quantum mechanics giving rise to a well defined group velocity. Transport faster than this group velocity is exponentially suppressed. Going back to the motivation of quantum state transfer, a natural comparison [37] for the time-scale of quantum state transfer is given by the time it would take to perform a sequence of swap gates (cf. Fig 1) that are realised by a pairwise switchable coupling Hamiltonian

[TABLE]

This time is linear in the chain length:

[TABLE]

Ideally one could say that the time for quantum state transfer should not take much longer than this. However one may argue that there is a trade-off between quick transfer on one hand, and minimising control on the other hand. A second cut-off time may be given by the decoherence time of the specific implementation. But short decoherence times could always be counteracted by increasing the chain coupling $J.$ A more general and implementation independent limit is given by the requirement that the peak width ${\Delta t}_{\mbox{peak}}$ should not be too small with respect to the total time. Otherwise it is difficult to pick up the state at the correct time. For the first peak, we can estimate the width by using the full width at half height of the airy function. From Eq. (28) we get an absolute peak width of $\Delta t_{\mbox{peak}}\approx 0.72N^{1/3}/J$ and a relative width of

[TABLE]

This is already quite demanding from an experimental perspective and we conclude that the transfer time should not be chosen much longer than those of the first peak.

4.6 How high should $p(t)$ be?

We have not discussed yet what the actual value of $p(t)$ should be to make such a spin chain useful as a device for quantum state transfer. $p(t)=0$ corresponds to no state transfer, $p(t)=1$ to a perfect state transfer. But what are the relevant scales for intermediate $p(t)$? In practice, the quantum transfer will suffer from additional external noise (Chapter 7) and also the quantum computer itself is likely to be very noisy. From this point of view, requiring $p(t)=1$ seems a bit too demanding.

From a theoretical perspective, it is interesting that for any $$