Controllable coupling of superconducting qubits and implementation of quantum gate protocols

Transcription

1 THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Controllable coupling of superconducting qubits and implementation of quantum gate protocols MARGARETA WALLQUIST Department of Microtechnology and Nanoscience Applied Quantum Physics Laboratory Chalmers University of Technology Göteborg, Sweden 2006

2 Controllable coupling of superconducting qubits and implementation of quantum gate protocols MARGARETA WALLQUIST ISBN c MARGARETA WALLQUIST, 2006 Doktorsavhandlingar vid Chalmers Tekniska Högskola, Ny serie Nr 2511 ISSN X Chalmers University of Technology Department of Microtechnology and Nanoscience (MC2) Applied Quantum Physics Laboratory SE Göteborg, Sweden Telephone +46 (0) ISSN Technical Report MC2-81 Chalmers Reproservice Göteborg, Sweden 2006

3 Controllable coupling of superconducting qubits and implementation of quantum gate protocols MARGARETA WALLQUIST Department of Microtechnology and Nanoscience Applied Quantum Physics Laboratory Chalmers University of Technology ABSTRACT The concept of a quantum computer was invented in the beginning of the 1980s as a quantum generalization of the reversible classical computer. After discoveries in the middle of the 1990s of quantum algorithms, which would solve some problems considered intractable for classical computers, the field of quantum computing has developed rapidly. A wide range of quantum systems are investigated for their possible ability to implement quantum algorithms in practice. The most promising solid state implementations are superconducting electrical circuits based on the Josephson effect, and the Coulomb blockade effect. Superconducting circuits are macroscopic quantum systems which can be fabricated using standard lithography technologies. Superconducting qubits - basic building blocks of a quantum computer - have been developed at several research laboratories, among others the MC2 here at Chalmers. This thesis presents a theoretical investigation of controllable coupling of superconducting qubits based on the single Cooper-pair box. Two coupling schemes are investigated in detail: current-controlled coupling via large Josephson junctions, and coupling via a superconducting stripline cavity. Both coupling designs are scalable and suitable for the charge regime of the single Cooper-pair box, as well as for the charge-phase regime where the qubit is better protected from environmental noise. Quantum gate protocols which are relevant for these physical couplings are discussed. The investigations show that, for both considered designs, the simplest two-qubit gate is the conditional phase-shift gate. This gate is universal for quantum computing, and it can create maximally entangled qubit pairs in only one run. The investigation combines methods from electrical engineering, quantum mechanics and quantum computer science. The work was supported by SSF Nanodev Consortium. Keywords: Quantum Computer, Quantum Electrical Circuit, Superconducting Circuit, Josephson Effect, Single Cooper-Pair Box, Qubit, Controllable Qubit-Coupling, Quantum Gate

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5 LIST OF PUBLICATIONS This thesis is based on the work contained in the following papers, I Josephson junction qubit network with current-controlled interaction J. Lantz, M. Wallquist, V.S. Shumeiko and G. Wendin, Phys. Rev. B 70, (R) (2004). II Current-controlled coupling of superconducting charge qubits M. Wallquist, J. Lantz, V.S. Shumeiko and G. Wendin, in Quantum Computation: solid state systems, Ed. P. Delsing, C. Granata, Y. Pashkin, B. Ruggiero and P. Silvestrini; Kluwer Academic Plenum Publishers (2004). III Superconducting qubit network with controllable nearest-neighbour coupling M. Wallquist, J. Lantz, V.S. Shumeiko and G. Wendin New J. Phys. 7 (2005) 178. IV Selective coupling of superconducting qubits via tunable stripline cavity M. Wallquist, V.S. Shumeiko and G. Wendin condmat/ (2006). Submitted to to Phys. Rev. B. v

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7 Contents 1 Introduction What is a quantum computer? Macroscopic quantum systems: the superconducting qubit Protocols for quantum computation The single Cooper-pair box qubit The simplest form of the SCB qubit The Josephson junction Deriving the classical circuit Hamiltonian Quantization and truncation The SCB qubit with a SQUID configuration The charge-phase regime Controllable coupling of single Cooper-pair box qubits Current-controlled coupling via large Josephson junctions Current-controlled coupling of two SCB qubits Current-controlled coupling via measurement junctions SCB asymmetry and other sources of residual interaction Maximum controllable interaction strength Experiment with fast gate voltage pulses - harmful or not? Scalability? A chain of N SCB qubits The charge-phase regime Qubit coupling via tunable transmission line cavity Dispersion equation of the tunable stripline cavity vii

8 viii Qubit coupling via an effective tunable LCoscillator Estimation of the cavity currents Nonlinear corrections Constraints on the circuit parameters The charge-phase regime Implementation of quantum gates Single qubit operations with SCB qubits Performing two-qubit gate operations using the current-controlled coupling of SCB qubits Two-qubit operations via the cavity Bell-pair construction Implementing the control-phase gate Discussion 61 Bibliography 69 Papers I VI 75

9 Was immer du tun kannst oder träumst es tun zu können, fang damit an! Mut hat Genie, Kraft und Zauber in sich. Johann Wolfgang von Goethe. ix

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11 Chapter 1 Introduction 1.1 What is a quantum computer? One day it will become technologically possible to build quantum computers, David Deutsch [1] stated in 1985 in an article titled Quantum computers and the Church-Turing principle. Deutsch followed a path set out by Richard Feynman a few years earlier. As Feynman pointed out, a classical computer is fundamentally not able to simulate the dynamics of a quantum system. Such a simulator should be realized with quantum elements obeying the quantum mechanical laws: nature isn t classical, dammit, and if you want to make a simulation of nature, you d better make it quantum mechanical [2]. Deutsch discovered a quantum generalization of the Turing machine - a universal quantum computer. Such a machine would be powerful enough to accomplish the task formulated by Feynman - efficient simulation of arbitrary quantum systems. Further breakthroughs in quantum computation came with the discoveries of quantum algorithms for solving particular mathematical problems not solvable with usual, classical computers. The first remarkable example was given by Deutsch and Jozsa [3]: the Deutsch-Jozsa algorithm is able to determine a global property of a function using only one function evaluation, while a classical computer will typically need an exponentially large number of evaluations. Other examples are the Grover algorithm for searching through an unsorted database [4], and the Shor algorithm for computing the discrete logarithm and for factorizing integers [5]. The Shor algorithm is an efficient quantum procedure for solving the problems which are considered intractable for classical computers. Not only does the discovery of this algorithm prove that quantum computers can outperform classical com- 1

12 2 Chap. 1: Introduction puters, but it also has a very important practical aspect - it threatens the ubiquitous cryptosystems used every day, for example for bank transactions. These cryptosystems rely on that no computer can factorize huge integers fast enough 1. The property of the quantum computer to have computational power exceeding that of a classical computer is of a great practical importance. Computations and simulations are important both in science, public services and industry. Simulations may provide information which is not achievable from experiments. Experiments are often more expensive and time-consuming; sometimes experiments are practically impossible to perform and computer simulations are a necessity in order to make predictions about a system. Weather forecast and traffic planning are just two of the numerous examples. Also, with the present development within nanotechnology, there is an increasing need to simulate the dynamics of systems on a quantum level. The simulation of an n-component quantum system would need a quantum computer with c n qubits (c is an integer), whereas a classical computer would need at least c n bits of memory for the same task [6]. That simulations of quantum systems grow fast with the size of the system is already a problem in e.g. quantum chemistry; classical computers are not large or fast enough to simulate the behaviour of the large molecules occuring in important biological systems. A major question at this point is, What device can implement the universal quantum computer that Deutsch suggested? The fundamental problem is that quantum states are fragile, and this fragility is a great challenge to anyone who tries to construct a quantum computer. The physical quantum computer is a many-body system, which, ideally, is to coherently evolve in time. However, in reality, the computer environment induces decoherence into the system. Especially dangerous in this respect are the control and readout devices: the major difficulty is to operate the system, and read out the information without destroying quantum coherence of the quantum computer. Sufficiently long life time of the quantum coherent state is a major requirement among the criteria formulated by David DiVincenzo [7] that any realistic quantum computer hardware is to obey. The other criteria are: the system must be scalable with well characterized elementary blocks - quantum bits, it should be possible to initialize the quantum bits in a specified state, 1 Luckely, the development within quantum cryptography, where the code is protected from attacks by the laws of quantum mechanics, is faster than the development within quantum computing. Quantum cryptosystems are already commercially available at e.g. the company idquantique in Geneva.

13 3 and to perform a universal set of operations - quantum gates, and to measure the state of the quantum bits. No single known physical system satisfies all these criteria. Atomic systems, for example, have very long coherence times, however, it is difficult to control them, and to make them scalable. On the other hand, scalable solid state quantum devices are vulnerable to environmental noise, coming from a huge number of microscopic degrees of freedom in the solid. The basic element of the quantum computer is a qubit - a quantum twolevel system characterized by the state Ψ, which is a superposition of the two computational basis states 0 and 1, Ψ = cos θ eiφ sin θ 1. (1.1) 2 Among the first ideas concerning qubit hardware was a theoretical proposal from 1995 to use the two electronic levels in barium ions [8]. The same year, quantum logic gates with electronic and vibrational levels of a beryllium ion were experimentally demonstrated [9]. Tremendous progress towards quantum computation was made by using nuclear spins in molecules as qubits manipulated with nuclear magnetic resonance (NMR) techniques. In 2001 the number 15 was successfully factorized using this kind of quantum computer [10]. All these qubits belong to the group of microscopic, natural qubits; natural in the sense that they are designed by the nature. 1.2 Macroscopic quantum systems: the superconducting qubit A quite different group of qubits are based on macroscopic quantum systems. For these devices, the qubit space is spanned by two macroscopically distinct states, which are collective states of a large number of microscopic particles. It was first pointed out by Leggett [11] that the quantum coherence could be observed in a macroscopic system - a superconducting ring interrupted by a Josephson tunnel junction (superconducting quantum interfererence device, SQUID). An illustrative example of a macroscopic qubit is the so called flux qubit, where the computational basis is defined by the clockwise and anticlockwise persistent currents in the superconducting ring of a SQUID [12, 13]. All internal degrees of freedom are frozen in the superconducting state. The only degree of freedom of the condensate of the Cooper pairs is the superconducting phase, which behaves as a single quantum particle. In the case of the flux qubit, each direction of the circulating current corresponds to

14 4 Chap. 1: Introduction a distinct phase state. The currents are clearly macroscopic objects, involving millions of Cooper-pairs. Macroscopic quantum systems are intermediate between microscopic quantum systems and macroscopic classical systems such as books and bikes; these systems exhibit quantum phenomena (tunneling for example), but unlike the microscopic particles, the macroscopic quantum systems are designed by humans for a given purpose, and fabricated according to these specifications. At first glance it may seem odd that macroscopic systems can exhibit quantum behaviour. Indeed, quantum mechanics originally described phenomena related to microscopic particles (atoms, electrons, photons), and the theory is still primarily associated with microscopic systems. It is therefore not intuitively clear how quantum physics can be associated with macroscopic variables such as the currents or voltages in an electrical circuit. Non-stationary processes in the electrical circuits are described by the equations of motion for currents and voltages, which are the Kirchhoff equations. The Kirchhoff equations are classical equations, but in fact, the underlying principle must be the quantum mechanical. The motivation for this statement is the quantum nature of the electromagnetic fields generated by the circuit dynamics. Since electromagnetic fields are quantized, there must exist a quantum generalization of the circuit equations of motion. Quantum effects are difficult to observe in normal electrical circuits. In a normal piece of wire, the quantum effects are washed out due to internal dissipation. In LC-circuits with high quality factor (i.e. low internal dissipation) the quantum behaviour could in principle be observed. Indeed, for an LC-oscillator in the THz regime the energy level spacing is of the order 10 K, and since it is possible to reach a temperature well below 1 K with a dilution refrigerator, the thermal mixing of the states can be made negligibly small. Dissipation broadens the resonance, but a high quality factor assures that the level width is much smaller than the inter-level spacing. Still no quantum effects are observed. The reason is that the expectation values of the harmonic oscillator observables follow the classical time evolution. This is a consequence of the Ehrenfest theorem. Thus, a quantum oscillator is distinguished from a classical oscillator only through the higher moments of the observables, which occur for example in the voltage and current fluctuations, and which are much more difficult to measure. The story is different for non-linear electrical circuits. An electrical circuit containing a non-linear element, such as a Josephson junction, displays quantum phenomena such as macroscopic quantum tunneling (MQT) [14]. However, since the dissipation may kill the quantum effects [15], weak dissipation is a necessary requirement besides the non-linear element. The experimental

15 5 discovery that a superconducting circuit can tunnel out of the superconducting state [16] indeed proved that the dynamics of a macroscopic degree of freedom - the phase difference across a Josephson junction - is governed by the laws of quantum mechanics. The quantum behaviour of the circuit was further confirmed by an experimental observation of the quantization of the energy levels in the Josephson junction potential [17]. The superconducting qubit is a macroscopic quantum system, with the two lowest energy levels being practically isolated from all other energy levels of the system. Quantum superpositions and coherent oscillations must exist only within this subspace. Circuit designs with a highly non-equidistant spectrum, which allows a truncation to a two-dimensional Hilbert space, were developed and experimentally tested by the end of the 90 s. Successful results were obtained for superconducting rings (SQUIDs) with distinct current states forming the computational basis (flux qubits) [12, 13], and for circuits based on the Coulomb blockade effect with a small superconducting island with distinct charge states forming the computational basis (charge qubits) [18, 19, 20]. A great advantage of the superconducting qubits is that the circuits are fabricated using standard lithography techniques, hence the technology is scalable. By modulating the biasing currents and voltages, it is easy to prepare the qubits in a given initial state. Various techniques for the read-out of individual superconducting qubits have been developed, including projective read-out techniques, for a review see [21, 22]. An important ingredient of the universal set of quantum gates - the control-not gate - was experimentally realized with coupled charge qubits in 2003 [23]. Thus the superconducting qubits very well meet the DiVincenzo criteria. However, the problem of insufficiently long coherence time remains the major problem to be solved. Another important problem to be solved concerns implementation of selective controllable qubit coupling: the interaction between the qubits must be switched on and off while performing the quantum gates. A theoretical investigation of this problem constitutes the central topic of this thesis. While discussing the interaction of the superconducting qubits, we restrict ourselves with the coherent qubit dynamics neglecting the decoherence effects. Meanwhile, one must keep in mind that decoherence properties are important for superconducting qubits. In fact, several decoherence mechanisms have been discovered, which give rise to quite short coherence time of superconducting qubits. An important source of losses is the radiation of electromagnetic waves into the circuit leads: due to rather high qubit frequencies (several GHz) the qubit wires behave as transmission lines. The dissipation in the leads is treated within the Caldeira and Leggett model

16 6 Chap. 1: Introduction [15], which considers the interaction of the qubit with a bosonic bath. The coherent evolution of a macroscopic two-level system under the influence of dissipation was discussed in [24]. Recent investigations have shown that decoherence also can arise from the interaction with a bath of two-level fluctuators in the bulk dielectrics and in the tunnel junctions [25]. The relevance of these dissipation sources is clear from the recent experiment [26] which shows that the relaxation time of a large-capacitance qubit can be increased by a factor 20 by reconsidering the material choice and the circuit design. 1.3 Protocols for quantum computation The implementation of a quantum algorithm consists of preparing the n- qubit state of a quantum computer in a given initial state, controlling the unitary evolution of the state according to a given pulse scheme, and in the end reading out the final state. A general quantum computation assumes the possibility to perform quantum operations with many interacting qubits (many-qubit gates). In practice, it may be extremely difficult to realize a many-qubit gate: for instance the hardware could only provide nearestneighbour coupling or a fixed coupling strength. To overcome this difficulty, a concept of universal gates, which is known for classical computers, is generalized for a quantum computer: a set of gates is defined which can be combined to implement any n-qubit operation. The quantum computer differs from the ordinary classical computer in one important aspect: the quantum computer operations are reversible, since the inverse of a unitary operation by definition always exists. A consequence is that no information is lost during gate operations; the number of output bits is always equal to the number of input bits. The inspiration for a universal set of quantum gates was therefore rather the reversible classical computer, a model for a computer which is logically reversible, and operates by purely classical means [27]. The three-bit Toffoli gate is a universal gate for the reversible classical computer [28], and indeed, in 1989 it was found that a three-qubit operation is a universal gate for quantum computation [29]. Further investigations revealed that the universal three-qubit gate can be implemented using only two-qubit and single-qubit gates [30]. This result relaxed the constraints on allowed hardware. Intuitively one understands that it is easier to realize controllable two-qubit interaction than the simultaneous coupling of three qubits. Still the two-qubit gates are the bottleneck here, because generally the evolution of individual qubits is more easy to

17 7 control, and requires a shorter gate operation time. What two-qubit gates are universal? Barenco found one family of universal two-qubit gates [31]; this result was further generalized to include almost any two-qubit gate [32]. A different point of view was provided by Barenco et al. [33], who studied the two-qubit control-not (CNOT) gate, which is the quantum analogue of the reversible exclusive-or gate. The CNOT flips qubit 2 only if qubit 1 is in the state 1, and it forms a universal set of gates together with single qubit rotations. This is remarkable since the exclusive-or gate is not universal for classical computation. It is easy to see that CNOT is an entangling gate. It transforms an input state such as ( ) 0 into a maximally entangled state, ( ). The most simple quantum computer consists of two qubits. Initially it is prepared in an input state Ψ(0), Ψ(0) = a 00 (0) 00 + a 01 (0) 01 + a 10 (0) 10 + a 11 (0) 11. (1.2) A gate operation, i.e. a unitary evolution U(t 0, 0), Ψ(t 0 ) = U(t 0, 0) Ψ(0), [ U(t 0, 0) = exp ī t0 ] dτh(τ), (1.3) h 0 is governed by the system Hamiltonian H, which generally has the form (in the qubit eigenbasis), H = i σ zi + κ mn σ m1 σ n2, (1.4) i=1,2 m,n=x,y,z where i is the energy difference between the two eigenstates of qubit i, and κ mn determines the strength and the symmetry of the two-qubit interaction. The Hamiltonian is here written in terms of the Pauli matrices, σ x = ( ), σ y = ( 0 i i 0 ), σ z = ( ), (1.5) using vector notation for the computational basis, ( ) ( ) 1 0 0, 1. (1.6) 0 1 Generally, the two-qubit interaction can result in a universal gate (provided the pulse time is chosen correctly), but most often this gate is not the CNOT gate. For example, the genuine two-qubit gate for interaction on the Heisenberg form, H = E ( ) σ 1 4 x σx 2 + σyσ 1 y 2 + σzσ 1 z 2, (1.7)

18 8 Chap. 1: Introduction is a SWAP gate [34], which creates linear superpositions of the input states 01 and 10, and must be applied twice in combination with single qubit rotations to produce a CNOT gate. Another example is the XY-interaction, H = E 4 ( σ 1 x σ 2 x + σ 1 yσ 2 y ), (1.8) where the genuine gate is an iswap gate [35], which takes 01 to i 10 and 10 to i 01. Also this gate is needed twice to make a CNOT. In fact, the focus on the CNOT gate is not particularly important, since any twoqubit gate, which is not a combination of single qubit gates and the SWAP 2, is universal [36]. A more relevant question is, what is the most efficient decomposition (into single- and two-qubit gates) of an algorithm, and how is this decomposition related to the genuine gate of my physical realization of the quantum computer? Another aspect of the efficiency of physical realizations for quantum computation, is the actual pulse time required to implement a useful gate operation. This is especially important when the qubits are non-resonant, 1 2. A diagonal Hamiltonian (all coupling constants are zero except of κ zz 0), produces a diagonal gate operation (the CPHASE gate), and in this case the pulse timing depends only on the coupling strength, T op h/κ zz. A physical realization which provides diagonal qubit coupling will be discussed in detail in this thesis. A transverse coupling, on the other hand, is sensitive to the qubit energy differences. For example, the gate operation time for an xx-coupling, κ xx 0, is enhanced by a factor ( 1 2 )/κ xx. Recent proposals to circumvent this problem, suggest to use simultaneous single qubit control pulses to effectively tune the qubits into resonance with each other [37] or use a dynamic control of the coupling element to get a similar effect [38]. With this trick, it is possible to achieve an effective tunable coupling from a weak, constant interaction. The Hamiltonian in Eq. (1.4) is written assuming a direct qubit coupling, i.e. that the coupling element degree of freedom can be integrated out. However, in some cases the coupling element is a dynamic object with a life of its own. In such a case, the pulse sequence of the gate operation must explicitly be made such that the coupling element returns to its ground state at the end of the calculation. Coupling via a bus is scalable if the bus only interacts with one qubit at a time. This is accomplished by separating the qubits in space (for example ions in ion traps [8]) or in time (for example atoms passing through QED cavities [39]) or in energy. In the latter case, 2 The SWAP gate only swaps the state of the two qubits, i.e Hence it is not an entangling gate.

19 9 one takes advantage of the qubit differences which are natural for fabricated qubits. The implementation of gate protocols with energy separated qubits will also be discussed in detail in the thesis. It should be stressed that the genuine gate of a physical quantum computer realization does not in the first place depend on the physical nature of the qubits, but rather on the symmetry of the interaction Hamiltonian. Therefore, quite different physical systems (such as ion traps vs. atoms in QED cavities) can exhibit the same symmetry of the qubit coupling and hence provide equivalent genuine gates. In this thesis we discuss protocols for the two-qubit gates, which are consistent with realistic physical couplings of the superconducting qubits.

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21 Chapter 2 The single Cooper-pair box qubit The superconducting qubits considered in this thesis include, as the central element, the single Cooper-pair box (SCB). The SCB is a small superconducting island with a controllable number of individual Cooper-pairs on it, connected to a superconducting reservoir through a Josephson tunnel junction [40]. The number of excess Cooper-pairs on the island is controlled by a voltage bias, applied via a gate capacitance to the island. To only allow the transfer of single Cooper-pairs, it is important that the island is in the Coloumb blockade regime, where the tunnel resistance is larger than the quantum resistance R Q = 25.8 kω [41]. This is equivalent to demanding that the energy of charging the island with a Cooper-pair, E C, is larger than the broadening of the island energy levels due to the tunneling. Moreover, the tunneling of individual Cooper-pairs is only possible to control if the energy of thermal fluctuations is smaller than the charging energy, k B T E C. An essential feature of the SCB is the ban on the odd electron number occupation of the island; this is achieved by assuming the superconducting gap to be larger than the charging energy of a single electron, > E C /4 [42]. The SCB can be employed as a qubit [18] - a basic building block for storing and manipulating one bit of quantum information [6]. The qubit dynamics is governed by the charging energy E C and the Josephson (tunneling) energy E J (for a review see [21]). The qubit can be operated in the charge regime, E J E C, where quantum information is stored in the number of excess Cooper-pairs, n = 0 or n = 1, on the SCB island. Two junctions in parallel (a low-capacitance SQUID) can substitute for the single junction, with the advantage that the coupling of the island to the reservoir can be controlled. This option has been experimentally implemented [20, 43]. A 11

22 12 Chap. 2: The single Cooper-pair box qubit modified version of the SCB with E J E C (the charge-phase regime), was also experimentally studied. In this parameter regime, the SCB qubit was found to be less sensitive to charge noise [44]. The energy eigenstates of the island form a computational basis in this regime. 2.1 The simplest form of the SCB qubit The simplest form of the SCB is a superconducting island coupled via a single Josephson junction to the reservoir of Cooper pairs. The circuit model is depicted in Fig The induced charge 2en g on the island is tuned using the gate voltage, V g, which is connected to the island via a gate capacitance C g : 2en g = C g V g. The gate voltage is used as a control knob for qubit manipulations. qubit island n C g φ Vg C E J Figure 2.1: A SCB qubit. Quantum information is stored in the number of excess Cooper-pairs, n = 0 or n = 1, on the island. n is conjugate variable to the phase φ across the Josephson junction. The Josephson junction is characterized by the capacitance C and Josephson energy E J. The dynamics of an electrical circuit is given by the Kirchhoff equations. However, the circuit dynamics can also be described by a Lagrangian form. This is an interesting fact, since the Lagrangian form was developed as an efficient way to describe mechanical systems [45]. The analogy between mechanical systems and electrical circuits is obvious when considering the LCoscillator, which is equivalent to the mechanical linear oscillator. Another example is the Josephson junction, the electrical analogy of a mechanical pendulum, with the junction capacitance corresponding to the pendulum mass [46]. The starting point for the formal description of the superconducting circuit will be the Lagrangian formalism for electrical circuits [47, 48, 49, 22].

23 13 When describing a superconducting circuit, a convenient choice for the generalized coordinate is the superconducting phase φ, related to the voltage V via the relation, φ = 2eV/ h. With this choice, the inductive energy plays the role of the potential energy. For example, the Josephson tunneling energy E J cosφ [50] acts as the potential energy of a Josephson junction. The capacitive energy of the junction has the form, CV 2 /2 = ( h/2e) 2 C φ 2 /2, and it plays the role of the kinetic energy The Josephson junction The Josephson effect was predicted by Brian D. Josephson in 1962 and it consists of dissipativeless flow of electrical current (supercurrent) through a tunnel junction with superconducting electrodes [51]. The superconducting condensate of the Cooper-pairs in the superconducting electrodes is described by a single macroscopic wave function, here denoted by ψ L = ψ L e iγ L at the left hand side and ψ R = ψ R e iγ R at the right hand side of the junction, respectively. The Josephson current is related to the phase of this wave ψ L ~ e iγ L ψ R ~ e iγ R C E J Figure 2.2: The Josephson tunnel junction, characterized by a capacitance C and a Josephson energy E J. The phase drop γ L γ R determines the current and the voltage drop across the junction. function via the following relation (Josephson relation), I = I c sin(γ L γ R ), (2.1) where I c = (2e/ h)e J is the maximum supercurrent that the junction can support. In the presence of a magnetic field, the phase jump across the junction, γ L γ R, is substituted for by the gauge invariant phase difference φ [50], φ = γ L γ R 2e L A d s, (2.2) h R where the magnetic vector potential A is integrated across the junction. A constant supercurrent I < I c flows through the junction without any voltage

24 14 Chap. 2: The single Cooper-pair box qubit drop. When voltage is applied to the junction, the phase φ becomes time dependent according to the Josephson relation [51], V = h dφ 2e dt. (2.3) To derive a Lagrangian for the Josephson junction, we need to find the φ E J Figure 2.3: Equivalent circuit of the Josephson junction; a capacitance and an ideal Josephson tunnel element (a non-linear inductor) in parallel. φ is the gauge-invariant phase difference across the junction. C kinetic and potential energies. This can be done by considering the work carried out by a source to establish a phase difference φ and a voltage V across the junction, t0 V (t)i(t) dt. (2.4) Bearing in mind that the Josephson junction consists of a Josephson tunneling element and a capacitor in parallel, Fig. 2.3, we can present the current as the sum of two components I J and I C. With the expressions for the supercurrent, Eqs. (2.1) and (2.2), and the voltage, Eq. (2.3), at hand, it is straight-forward to evaluate the inductive energy, t0 V (t)i J (t) dt = h 2e I c φ sin φ dφ = E J (1 cosφ). (2.5) Similarly, the charging energy can be evaluated with help of the relation I C = C V, t0 V V (t)i C (t) dt = C V dv = CV 2 ( ) 2 h C = φ 2 2 2e 2. (2.6) The last equality holds also for capacitors, if we formally define the capacitor phase φ C as [48, 49], t0 2e φ C (t 0 ) = h V (t)dt.

25 15 Josephson tunnel junctions, used for qubit applications, are made with aluminum [20, 43] or niobium [52] superconducting electrodes separated by thin (a few nanometer) insulating layers (aluminum oxide) Deriving the classical circuit Hamiltonian Now let us write a Lagrangian for the SCB circuit, which is the difference between capacitive (kinetic) and inductive (potential) energies, L SCB = ( ) 2 ( ) 2 h C 2e 2 φ h 2 C g + 2e 2 ( ) 2eV 2 g φ + + E J cos φ. (2.7) h The second term is the energy of the gate capacitor, which is derived using the Kirchhoff voltage law. The Hamiltonian provides an equivalent formulation of the circuit dynamics. We need the circuit Hamiltonian in order to perform the canonical quantization procedure, which leads to a quantum description of the electrical circuit [47, 48, 49, 53]. With a Legendre transformation, a classical Hamiltonian H is derived from the Lagrangian function, H(n i, φ i ) = i ( hni φi ) L, ni = L h φ i, (2.8) using the last equality to let the conjugate momenta n i substitute for the corresponding phase velocities φ i. Note that φ i and n i are dimensionless quantities. The Hamiltonian could also be written in terms of fluxes Φ i = ( h/2e)φ i and charges Q i = 2en i, i.e., in this description charge and flux are conjugate variables. For the SCB, the conjugate momentum, n = [( h/2e)c Σ φ + C g V g ]/2e, turns out to be the (dimensionless) charge on the island, and C Σ = C + C g the total capacitance of the island. The explicit form for the (classical) Hamiltonian is, H SCB = E C (n n g ) 2 E J cosφ, (2.9) where the capacitive term is described by the charging energy of the box, E C = (2e) 2 /(2C Σ ) 1. 1 A constant term C g V 2 g /2 was dropped out, since it does not affect the system dynamics.

26 16 Chap. 2: The single Cooper-pair box qubit Quantization and truncation The Hamiltonian in Eq. (2.9) is a classical function of the dynamical variables φ and n, obeying the Poisson bracket relation [45], {φ, n} = 1. (2.10) The quantization procedure replaces the variables n and φ by quantum mechanical operators 2 ˆn and ˆφ obeying the canonical commutation relation [54, 48, 49], [ˆφ, ˆn ] = i, (2.11) thus the corresponding quantum Hamiltonian reads, H SCB = E C (ˆn n g ) 2 E J cos ˆφ. (2.12) Close to the edge of the Brillouin zone, n g = 1/2, the two charge states with zero and one excess Cooper-pair ( n = 0 and n = 1 ) are almost degenerate, as shown in Fig 2.4. The spectrum is split up by the Josephson term and the charge states are mixed. When the charging energy is much larger than the E E E J _ 2 n g 1/2 n= 1 n=0 n= n g Figure 2.4: Left: spectrum of the SCB when E J = 0, shown for n = 1, n = 0 and n = 1. The charge states with n = 0 and n = 1 are degenerate at n g = 1/2. Right: The degeneracy at n g = 1/2 is split up by the Josephson coupling which mixes the charge states. The eigenstates at the charge degeneracy point are 1/ 2( 0 ± 1. Josephson energy, the two lowest eigenstates - superpositions of 0 and 1 - are well separated from higher charge states. Thus it is possible to truncate the Hamiltonian to the subspace constituting these two states [18]. Before 2 Hats will be used to distinguish quantum operators only in this subsection.

27 17 truncation, we project the Hamiltonian onto the charge states n, H SCB = [ E C (n n g ) 2 E ( J e iˆφ + e )] iˆφ n n, ˆn n = n n. n 2 (2.13) To evaluate the effect of the Josephson term on the charge state n, we use the Baker-Hausdorff lemma as described in [54], ˆne iˆφ n = e iˆφˆn n i [ˆφ, ˆn ] e iˆφ n = (n + 1)e iˆφ n, (2.14) to see that e iˆφ n is an eigenstate of ˆn with eigenvalue n+1: e iˆφ n = n+1. Analogously, e iˆφ n = n 1. Thus the Josephson term couples different charge states and is clearly responsible for the tunneling of Cooper-pairs into or out from the island. The quantum Hamiltonian takes the form, H SCB = n [ E C (n n g ) 2 n n E J 2 ( n + 1 n + n 1 n ) ]. (2.15) The charging term of the Hamiltonian determines the energy needed to put another Cooper-pair on the SCB island, and it increases quadratically with n. When E J is weak, the only tunneling which is allowed by the large charging energy E C is between charge states with (almost) equal energy, such as 0 and 1. With a temperature low enough that excitation to higher charge states is prohibited, it is allowed to truncate the Hamiltonian to the subspace spanned by { 0, 1 }. Thus in the limit E J E C, T E C, the SCB Hamiltonian is reduced to the two-level system Hamiltonian H q, ( H q = E C n g 1 ) σ z E J 2 2 σ x, (2.16) here written in terms of Pauli spin matrices: σ z = and σ x = The SCB qubit with a SQUID configuration A disadvantage of the single junction SCB, is that only the electrostatic term is controllable, Eq. (2.16), through the gate voltage V g. The circuit can 3 The term proportional to the unity matrix, (1/2 n g (1 n g ))1, is dropped out since it does not contribute to the qubit dynamics.

28 18 Chap. 2: The single Cooper-pair box qubit however be further generalized - to gain control over the Josephson term - if the single Josephson junction is split into two junctions forming a SQUID loop, as depicted in Fig. 2.5 [21]. An applied magnetic flux Φ e will determine the total phase across the two junctions, hence control the effective Josephson energy of the circuit. To begin with, we generalize the Lagrangian Eq. (2.7) qubit island Φ e φ i C g Vg C E J Figure 2.5: A SCB qubit with a SQUID configuration to gain control over the Josephson term. of the simple SCB circuit to include both junctions and the inductive energy of the loop, L SCB = ( ) 2 [ h C ( φ2 2e φ ) ( 2 C g 2e h V g φ ) 2 ] 1 + E J (cosφ 1 + cosφ 2 ) Φ2 ind 2L loop, (2.17) here we have assumed equal Josephson junctions. In practice the deviation between different junctions is typically a few percent, even up to 10 %. Consequences of the SCB asymmetry will be discussed more in the next chapter in the section about current-controlled coupling. Because the circuit forms a loop, the flux quantization relation is applicable [50], i=1,2 φ i = 2π Φ 0 Φ. (2.18) where Φ 0 = hπ/e is the flux quantum. The flux quantization relation states that the sum of the junction phases must equal the total magnetic flux in the loop, Φ, which is a combination of the induced flux Φ ind and the externally

29 19 applied flux Φ e. Assuming that the self-inductance of the loop is very small, the induced flux Φ ind must go to zero, L loop 0 Φ ind = Φ 0 2π (φ 1 + φ 2 ) Φ e 0, (2.19) and the applied magnetic flux Φ e imposes a constraint on the two phases, thus leaving just one degree of freedom for the SQUID, φ = φ 2 φ 1. (2.20) 2 The two junctions behave effectively as a SCB with a single junction, L SCB = ( ) 2 [ h C 2e φ 2 + C ( g 2e 2 h V g + φ ) 2 ] + 2E J cos with island charge 2en, ( πφe Φ 0 ) cosφ, (2.21) 2en = 2e L h φ = hc Σ 2e φ + 2en g. (2.22) Here C Σ = 2C + C g is the total capacitance of the island 4. The result for the classical Hamiltonian reads, ( ) H SCB = E C (n n g ) 2 πφe 2E J cos cosφ. (2.23) Similarly to the single junction SCB case, the corresponding quantum Hamiltonian is projected onto the charge states n, [ E C (n n g ) 2 n n E J cos ( πφe Φ 0 ) ] ( n + 1 n + n 1 n ). H SCB = n Φ 0 (2.24) We truncate the Hamiltonian to the subspace of the two lowest charge states 0 and 1 (Fig. 2.4) and arrive at the two-level system Hamiltonian, ( H q = E C n g 1 ) σ z E J cos 2 ( πφe Φ 0 ) σ x. (2.25) The major qualitative difference from the Hamiltonian of the simpler SCB circuit, Eq. (2.16), is the possibility, provided by this circuit design, to control the Josephson term. This difference is important when it comes to performing quantum gates. 4 Compare with the single junction SCB circuit (Eq. (2.7)) for which C Σ = C + C g

30 20 Chap. 2: The single Cooper-pair box qubit At the charge degeneracy point, n g = 1/2, the qubit is less sensitive to charge noise, with an increasing coherence time as a result [44, 43, 57]. Assuming that the charge degeneracy point is chosen as the working point during both single- and two-qubit gates, we write the Hamiltonian in the eigenbasis at the charge degeneracy point, ± = 1 2 ( 0 ± 1 ), (2.26) which is connected to the charge basis { 0, 1 } by a Hadamard transformation (Eq. (4.4)). In this basis the Hamiltonian reads, ( ) πφe H q = E C δn g σ x E J cos σ z, (2.27) where we wrote δn g = n g 1/2 to indicate that this basis is most suitable when we are close to n g = 1/2. An interesting question is - are there currents in the loop, and if so - how large are they and how are the current eigenstates related to the energy eigenstates? Consider both junctions as a single junction with phase φ + = φ 1 + φ 2, forgetting the flux quantization relation for a moment. The loop current I is the time derivative of the charge on this single junction, 2en +, which is the conjugate momentum of φ +. We use the Hamilton equations of motion to calculate the current; Î = 2eṅ + = 2e H SCB h φ + = 2e E J h 2 sin ( ) φ+ 2 Φ 0 σ z = I ( ) c 2 sin πφe σ z. (2.28) Φ 0 Since the current operator is diagonal (σ z ), the energy eigenstates ± at the charge degeneracy point are also eigenstates of the current circulating in the loop. At zero bias flux, Φ e = 0, the eigenstates are separated by the energy splitting 2E J, but there are no circulating currents. Actually it could not be otherwise, because for zero flux there is a symmetry between clockwise and anti-clockwise rotation in the loop. Non-zero currents would indicate that one direction (clockwise or anti-clockwise) is energetically favorable. An asymmetry is created when the bias flux is applied, and the sign of the flux determines which direction should be favored. A consequence of the fact, that the current operator is diagonal in the qubit basis, is that current measurements can be used for qubit non-demolition read-out. This is discussed in [44, 55] and also later in this thesis in relation to the controllable coupling.

31 The charge-phase regime We previously considered the charge regime for the SCB island, E J E C, where the two lowest charge eigenstates, n = 0 and n = 1, constitute the qubit basis. In this regime the gap between the energy eigenstates at n g = 1/2 is equal to the Josephson energy, hence very small, and the derivative of the energy band changes abruptly, as we can see in Fig 2.4. Any fluctuation in the gate voltage creates an unwanted variation in the qubit energy (except in a very close vicinity to n g = 1/2) and may cause dephasing. In the charge-phase regime, E J E C, the gap is wider, and the energy bands are smoother. Hence fluctuations of the gate voltage affect the qubit energy less than in the charge regime, and the sensitivity to charge noise is smaller [44]. Due to the relatively large Josephson term (E J /2)( n + 1 n + n 1 n ) of the SCB Hamiltonian (Eq. (2.15)), tunneling between several charge states is allowed and the truncation to the two charge states 0 and 1 is not possible. The two lowest eigenstates are given by Bloch wave functions, which consist of superpositions of several charge states [56]. We truncate the Hamiltonian to the two lowest eigenstates E + and E near the working point n g = 1/2, where the first derivative of the energy disappears. The disappearance of the first derivative is advantageous from the decoherence point of view, since gate voltage fluctuations only cause second-order variations in the SCB energy [44, 57]. The resulting Hamiltonian is, H SCB H q = σ z + λe C (n g 1/2)σ x, = E + E. (2.29) It is clear that charge cannot serve as a computational basis in this case. The choice of computational basis is, however, by no means unique [21], and whether a certain basis is convenient or not depends on e.g. the measurement basis of the read-out device. The quantum capacitance read-out device is suitable for the SCB qubit in the charge-phase regime [58, 59]. This device measures the second derivative of the energy band at n g = 1/2. A convenient qubit basis for this device should therefore be the energy eigenstates at n g = 1/2, i.e. E + and E. A current measurement is another way to read out the loop-shaped SCB qubits. We saw in the previous section that this method provides quantum non-demolition measurement in the charge regime. A similar method has been employed for read-out in the charge-phase regime [44]. It is, however, not at all obvious that the non-demolition property persists in the chargephase regime, since it relies on that the current operator preserves its diagonal form when the qubit basis (the Bloch wave functions) change due to the increasing Josephson energy. For this investigation, let us consider the SCB

32 22 Chap. 2: The single Cooper-pair box qubit Hamiltonian (Eq. (2.24)) at n g = 1/2, H 1 = n= [ E C n(n 1) n n E J cos ( πφe Φ 0 ) ] ( n + 1 n + n 1 n ). (2.30) In order to say something about the current operator in the truncated basis, we must try to find out what the two lowest Bloch states E + and E are. It is reasonable to believe that the two lowest eigenstates in the charge regime, ± 0 ± 1 (Eq. (2.26)), will develop into the lowest Bloch states when the Josephson energy increases. Therefore a partition into positive charge states n > 0 and negative charge states n 0 could be clever. The states are labelled with mσ, σ =,, and 1 < m <, such that, and 1, 2, 3,... = 1, 2, 3,... 0, 1, 2,... = 1, 2, 3,... In Eq. (2.26), the energy eigenstates (at n g = 1/2) ± are connected to the charge states via a Hadamard transformation, Eq Similarly, we perform a Hadamard transformation in σ-space, which takes the charge states mσ into the superpositions m± = (1/ 2)( m ± m ). Then we arrive at a Hamiltonian which is diagonal in σ-space, EC ẼJ ẼJ 2E C ẼJ ẼJ ẼJ H 1 =...., (2.31) EC ẼJ ẼJ 2E C ẼJ ẼJ here ẼJ = E J cos(πφ e /Φ 0 ). From this form it is clear that the two lowest diagonal entries of each block, marked with X in the following sketch, X X, Ẽ J

33 23 are the two lowest eigenstates when the Josephson energy E J goes to zero. When E J increases, the eigenstates develop into superpositions of the states within the same block. Also, the elements marked X will always be the lowest eigenstates. This follows from the fact that the Schrödinger equation of the SCB is a Mathieu equation, whose solutions do not cross when the potential (E J ) increases [60]. Thus, the procedure is to diagonalize each block ({ m+ } and { m }, respectively) independently and then truncate the Hamiltonian to the two lowest eigenstates E + and E, i.e. the two states marked X. The operator of the SCB loop current, Î ( H/ φ + ), which is derived from the SCB Hamiltonian H, Eq. (2.24), has the operator form n + 1 n + n 1 n before truncation. In order to truncate the current operator to the qubit basis, we follow the steps of the SCB Hamiltonian truncation above. In the first step, which is to write the operator in the m± -basis, we immediatlely see that the current operator aquires a blockdiagonal form, I (2.32) The second step, the rotations which diagonalize the Hamiltonian (Eq. (2.31)), preserves the block-diagonal form of the current operator, because the rotations only mix states within the blocks. Because of its block-diagonal form, the current operator can not mix SCB states belonging to different blocks. In particular, it cannot mix the two eigenstates of the charge-phase qubit. Hence, in the qubit basis the current operator is diagonal, Î = I 0 σ z. The conclusion is that the qubit states are eigenstates of the truncated current operator, and loop current measurement is suitable for non-demolition qubit read-out. Another question which should to be addressed is whether we can still couple the qubit states by departing from n g = 1/2, i.e. if the second term of Eq. (2.29) exists as it is written. For this analysis, we consider the terms of the SCB Hamiltonian which are proportional to the departure from n g = 1/2,