25.2.2.1 Ising and Transverse (XY) Coupling

In semiconductor optical cavities, the coupling between spins in QDs can be achieved without tunnel coupling between the dots, but instead via emission and reabsorption of virtual cavity photons (16) (see Section 25.3.3). In this case, the exchange coupling between the electron spins is no longer described by the isotropic exchange Hamiltonian equation 25.1, but by the XY (transverse) spin Hamiltonian,

25.12

where we chose the basis of the two interacting qubits and for the matrix representation of . In inductively coupled SC qubits ( 5, 11) (Section 25.4), the coupling also has the XY form, Eq. 25.12; thus, it is of interest whether this coupling can be used for universal quantum computing instead of the isotropic Heisenberg Hamiltonian.

In this context one should point out that any generic two‐qubit Hamiltonian gives rise to a universal set of gates when combined with single‐qubit operations. Here, we discuss how a universal gate ( , ) can be constructed explicitly from anisotropic exchange interaction. In two notable cases of anisotropic spin couplings, the Ising and the XY interactions, it is known how the and gates can be constructed. In the case of a system described by the Ising Hamiltonian and a homogeneous magnetic field in the ‐direction, there is a particularly simple realization of the gate with constant parameters, namely (2). However, a pure Ising coupling is rarely found in nature (although it has to be said that motional narrowing in liquid‐state NMR leads to an approximately pure Ising coupling).

For the transverse spin–spin coupling of Eq. 25.12, we find that a useful two‐qubit gate, such as the CPF operation, can be done by combining with one‐bit rotations. The unitary evolution operator generated by the Hamiltonian of Eq. 25.12 is

25.13

where . The CPF gate ( ) can be realized by the sequence ( 16,17)

25.14

where denotes the vector Pauli operator, , and and . The gate can be realized by combining the CPF operation with single‐qubit rotations as in Eqs. 25.6 and 25.7.

While it is impossible to generate the CNOT gate with a single use of the XY Hamiltonian (17), it is possible to generate a different universal quantum gate with the XY interaction in a single pulse; the CNOT + SWAP (CNS) gate , is generated as (18)

25.15

where is the Hadamard gate applied to qubit .

25.2.2.2 Anisotropy Due to the Spin–Orbit Coupling

Even in the case of spins occupying tunnel‐coupled sites (such as QDs), where the exchange is described by the isotropic Hamiltonian, Eq. 25.1, the isotropy can be broken due to spin–orbit coupling during tunneling between the sites (see (9) and references therein). Surprisingly, it turns out that the first‐order effect of the spin–orbit coupling during quantum gate operations can be eliminated using time‐symmetric pulse shapes for the coupling between the spins. A related, but independent, result shows that the spin–orbit effects exactly cancel in the gate sequence on the right‐hand side of Eq. 25.5 required to produce the quantum XOR gate, provided that the pulse form for the spin–orbit and the exchange couplings are identical. The XOR gate being universal when complemented with single‐qubit operations, this result implies that the spin–orbit coupling can be dealt with in any quantum computation. In any real implementation, there will be some (small) discrepancy between the pulse shapes for the exchange and the spin–orbit coupling; however, one can choose two pulse shapes that are very similar. It was shown that the cancellation still holds to a very good approximation in such a case, that is, the effect of the spin–orbit coupling will still be strongly suppressed.

We will now discuss the cancellation of the spin–orbit effects in the sequence equation 25.5 required for the XOR gate in detail. The spin–orbit coupling for a conduction‐band electron is given by the following Hamiltonian, being linear in the 2D momentum operator , ([100] orientation of the 2D plane),

25.16

where the constants depend on the strength of the confinement in the ‐direction and are of the order for GaAs heterostructures. Combining the isotropic Heisenberg coupling 25.1 with the anisotropic exchange between two localized spins and , one obtains the spin Hamiltonian

25.17

where the anisotropic part is given by the expression

25.18

and is the spin–orbit field, the ground state in site (QD) , and . As discussed in Section 25.5, for , the quantum XOR gate can be obtained by applying twice, together with single‐spin rotations, see Eqs. 25.5 and 25.7. Moreover, if , then commutes with itself at different times, and the time‐ordered exponential

25.19

is a function of the integrated interaction strength only, . In particular, is the “square root of SWAP” gate.

Let us consider now the more interesting situation where . If in this case, and (and thus ) are time independent, then still commutes with itself at different times, and one can find a fixed coordinate system in which is parallel to the ‐axis. In this basis, the anisotropic term Eq. 25.18 can be expressed as

25.20

with . In the singlet–triplet basis of the two spins with basis vectors the gate sequence Eq. 25.5, including the anisotropy Eq. 25.18, produces the unitary operation

25.21

where denotes the diagonal matrix with diagonal entries . The pulse strength and the spin–orbit parameters only enter in the subspaces, while the terms linear in have canceled out exactly in . By choosing , one obtains the CPF gate , equivalent to the XOR via the basis change Eq. 25.7. In conclusion, we have shown that the anisotropic terms . in the spin Hamiltonian cancel exactly in the gate sequence equation 25.6 for the quantum XOR. In real systems, we can expect that the anisotropic terms in the Hamiltonian are not exactly proportional to , that is, that is time dependent. It can be shown that, nevertheless, for small deviations from proportionality, the cancellation described earlier still holds to a good approximation (e.g., within a reasonable error‐correction threshold).

25.2.4.1 Encoding

The smallest number of spins 1/2 for which two orthogonal states with identical exist is three. The space of three‐spin states with spin quantum numbers , is two‐dimensional and will serve as our encoded qubit. We make the following explicit choice for the basis states of the qubit:

25.25

25.26

where is the singlet state of spins 1 and 2 of the three‐spin block, and and are triplet states of these two spins. In principle, this solves the problem of exchange‐only quantum computing, but in practice, we would like to know what the cost in terms of qubits (for coding) and gates (for operating on encoded qubits with the exchange interaction) will be, and explicitly how a universal set of operations on the encoded qubits can be achieved.

Universal quantum computing is also possible uniquely with the anisotropic XY interaction 25.12, a result which was later generalized to large class of anisotropic exchange Hamiltonians. An encoding involving two spins per qubit has also been demonstrated for universal quantum logic starting from locally alternating ‐factors and from a homogeneous magnetic field combined with anisotropic exchange interactions..

25.2.4.2 One‐Qubit Gates

Unitary gates on a single encoded qubit (a block of three spins) are performed as follows. The exchange between code qubits 1 and 2, , generates a rotation , which is a ‐axis rotation (in Bloch‐sphere notation) on the encoded qubit, while produces a rotation about an axis in the x–z‐plane, at an angle of 120° from the ‐axis. Since simultaneous application of and can generate a rotation around the ‐axis, three steps of exchange coupling suffice to implement any one‐qubit rotation using the classic Euler‐angle construction, assuming nearest‐neighbor coupling in a linear arrangement of the code block and allowing for parallel operations. In serial operation, that is, if each exchange coupling is switched on after all others have been turned off, it can be found numerically that four steps are always adequate when only nearest‐neighbor interactions are possible, while three steps suffice if interactions can be turned on between any pair of spins.

25.2.4.3 Two‐Qubit Gates

It is less straightforward to understand the implementation of a two‐qubit gate such as using the exchange interaction on two three‐spin code blocks. While the four basis states have total spin quantum numbers , , the complete space with these quantum numbers for six spins is nine‐dimensional. Numerical searches for the implementation of two‐qubit gates using a simple minimization algorithm have resulted in an apparently optimal sequence for an encoded ( ) operation comprising 19 exchange operations in series. (Variations of this result with other than linear arrangements of the constituent qubits and with parallel operation exist.)

25.2.4.4 Resonant Exchange, Hybrid, and Always‐On Exchange‐Only Qubits

A variety of three‐spin qubits sharing many features with the exchange‐only qubit described earlier have been put forward recently. Here, we only give a brief overview and refer the interested reader to a more detailed review (21). To allow for driving of the qubit with resonant oscillatory electric fields, the exchange coupling between all three QDs can remain on all the time, thus producing an energy splitting matching the frequency of the driving fields. This type of three‐spin qubit is known as resonant exchange (RX) qubit (22–24). While the always‐on exchange coupling enables electric control with resonant fields, it also opens new decoherence channels by coupling the spin qubit to electrical noise, see also Section 25.4.3. Another three‐spin qubit with partial charge character is the so‐called hybrid qubit, which consists of three electrons contained in two QDs (25,26). Reducing the number of QDs to one single dot hosting all three electrons, we arrive at the spin–charge qubit (27). Finally, we mention the always‐on exchange‐only (AEON) qubit, which is, as the RX qubit, a variant of the exchange‐only qubit with the exchange couplings always turned on, but operated in a different regime, which can under certain conditions be less susceptible to charge noise and offers new ways for two‐qubit coupling (28).

25.2.4.5 Spin Cluster Qubits

A different kind of encoded qubits, the so‐called spin cluster qubits, have been suggested in order to relax the requirements for control on the single‐spin level while inheriting the favorable single‐spin properties such as long decoherence time and fast gate operating time. Spin cluster qubits are finite spin chains with Heisenberg or anisotropic (XY and Ising‐like) antiferromagnetic exchange interaction ( ).

25.3.2.1 The Heitler–London Approach

The HL approximation has its origin in molecular physics: we can think of our double‐dot systems as a hydrogen molecule . In the HL approach, we start from single‐dot ground‐state ( wave) orbital wavefunctions and combine them into the (anti‐) symmetric two‐particle orbital state vectors

25.31

the positive (negative) sign corresponding to the spin singlet (triplet) state and denoting the overlap of the right and left orbitals. A nonvanishing overlap implies that the electrons tunnel between the dots (see also Section 25.3.2). Here, and denote the one‐particle orbitals centered at , and are two‐particle product states. The exchange energy is then obtained through . The single‐dot orbitals for harmonic confinement in two dimensions in a perpendicular magnetic field are the Fock–Darwin states, which are the usual harmonic oscillator states, magnetically compressed by a factor , where denotes the Larmor frequency. The ground state with energy , centered at the origin, is . Shifting the single‐particle orbitals to in the presence of a magnetic field, we obtain , where the phase factor involving the magnetic length is due to the gauge transformation . Splitting the Hamiltonian 25.28 according to , where is the single‐electron Hamiltonian of a parabolic QD at site , we obtain (29)

25.32

with the overlap integral . Evaluation of the matrix elements of and yields

25.33

where is the dimensionless distance and the zeroth order Bessel function. The first and second terms in Eq. 25.33 are due to the Coulomb interaction , where the exchange term enters with a minus sign. We have introduced the parameter ( , for meV) as the ratio between Coulomb and confining energy. The last term in 25.33 has its origin in the confinement potential . We plot the exchange coupling in Figure 25.5 (dashed line). As we have anticipated earlier, the ground state at is a singlet (thus ). However, we also see here that the singlet need not be the ground state for finite magnetic fields. In fact, in our example, changes sign from positive to negative at . This singlet–triplet crossing occurs over a wide range of parameters and . At ( ) and , the singlet–triplet crossing occurs at about . The transition from antiferromagnetic ( ) to ferromagnetic ( ) spin–spin coupling with increasing magnetic field is caused by the long‐range Coulomb interaction, in particular by the negative exchange term, the second term in Eq. 25.33. As ( for meV), the magnetic field compresses the orbits by a factor and thereby reduces the overlap of the wavefunctions, , exponentially strongly. Similarly, the overlap decays exponentially for large interdot distances, . There is a subtlety regarding this exponential suppression, however, namely that it is partly compensated by the exponentially growing exchange term . As a result, the exchange coupling decays exponentially as for large or , as shown in Figure 25.6b for ( ). What is important for quantum gate operations is that the exchange coupling can be tuned through zero and then suppressed to zero by a magnetic field in a very efficient way.

25.3.2.2 Limitations and Extensions of the Heitler–London Approach

The HL approximation breaks down explicitly (i.e., becomes negative even when ) for some values of the interdot distance if the interaction becomes too strong. For the choice of parameters made earlier, this happens as exceeds .

Several improvements of the HL method are possible – we discuss two such improvements that have been studied for the double QD case.

1. sp‐hybridization: The HL approximation can be improved by taking into account more than just the lowest single‐dot orbital. Admixture of higher orbitals can be taken into account using a variational approach; the orbitals obtained in this way are termed hybridized orbitals, in analogy to hybridized molecular orbitals in chemistry. We plot the result of a calculation using sp‐hybridized QD orbitals in Figure 25.5.
2. Hund–Mulliken approximation: The HL approximation is restricted to QDs that are occupied with a single electron. Even with a single orbital per site, the Pauli principle allows for the presence of a second electron with opposite spin on a QD orbital. This admixture of double occupancy is suppressed by the repulsive Coulomb interaction between electrons, thus justifying the HL approximation (to some degree). However, it is better to take the two doubly occupied states into account explicitly. This is done in the Hund–Mulliken (molecular orbit) approximation, which we discuss in the following section.

25.3.2.3 The Hund–Mulliken Approach and the Hubbard Limit

In the Hund–Mulliken (HM) or molecular orbit approximation, the HL approach is extended by also including the two doubly occupied states. Due to the Pauli principle, these additional states have to be spin singlets (29). In this manner, we have enlarged the orbital Hilbert space from two to four dimensions. In order to write down a HM model, we first need to orthonormalize the single‐particle states. This yields the states , where again denotes the overlap of with and . Diagonalizing

25.34

in the space spanned by , we obtain the eigenvalues , (singlet), and (triplet), where the quantities , , , and are given in (29). The exchange energy then becomes

25.35

For short‐range Coulomb interactions (and in the absence of a magnetic field), reduces to , where denotes the hopping matrix element and the on‐site repulsion. Thus, and are the generalized hopping matrix element and the on‐site repulsion in an extended Hubbard model, renormalized by long‐range Coulomb interactions. The remaining two singlet energies, and , are separated from and by a gap of order and are therefore neglected for the study of low‐energy properties. Typically, the “Hubbard ratio” is less than 1, for example, if , meV, and , we obtain , and this ratio decreases with increasing . Therefore, we are in an extended Hubbard limit, where takes the form

25.36

The first term in Eq. 25.36 corresponds to the standard Hubbard approximation but with and being renormalized by long‐range Coulomb interactions. The term is of long‐range Coulomb nature; it accounts for the difference in Coulomb energy between the singly occupied singlet and triplet states . It is the term that makes negative for high magnetic fields, whereas for all values of (see Figure 25.6a). Thus, the usual Hubbard approximation (i.e., without ) would not give reliable results, neither for the ‐dependence (Figure 25.6a) nor for the dependence on the interdot distance (Figure 25.6b).

25.3.2.4 Numerical Work

The calculations we have discussed so far take into account only the ground‐state orbital in each QD, with the exception of the sp‐hybridized HL, where two additional p‐orbitals are included. The HM can be refined by including a number of higher QD orbitals as well. Refined calculations of this type are usually done numerically and are very closely related to Hartree–Fock (HF) calculations. However, HF is not sufficient for the purpose of calculating a spin exchange coupling , since it is not capable of including entangled (quantum correlated) states such as the spin singlet or triplet. This is typically remedied by invoking the so‐called configuration‐interaction (CI) method, which includes linear superpositions of HF states. Numerical studies of the double‐dot system with one and three electrons per QD showed good agreement with the somewhat more crude approximations discussed earlier (9).

25.3.2.5 Measurements of Quantum Dot Exchange

Signatures of singlet–triplet crossings have been observed using transport spectroscopy in lateral GaAs QD structures (33) (see Figure 25.7). Although a single elongated dot structure was used, there are signatures that a double dot was formed in the experiment (7). These data seem to be in rather good qualitative agreement with theory (29), bearing in mind that the absolute magnitude of the exchange coupling strongly depends on the interdot distance, which is a free parameter of the theory. Similar double‐dot experiments with the double‐dot systems shown in Figure 25.2 are in preparation.

25.3.3.1 Optical Semiconductor Microcavities

The proposal for spin‐based quantum computation as presented in Section 25.3 makes use of the exchange coupling that arises when electrons are allowed to tunnel from one QD to the adjacent QD (2). Note that this scheme is universal for quantum computing despite the locality of the physical exchange interaction. In particular, arbitrary remote pairs of spins can be coupled using the exchange coupling to SWAP spins and bring two distant spins into proximity. There is a modification of this original scheme in which the proximity of two spins is not required even at a physical level, because the interaction is mediated by a resonant mode of an electromagnetic cavity (16). The control of this interaction, as well as single‐qubit operations, is achieved using focused laser fields applied to the QDs.

The scheme is based on doped QDs that are embedded in a semiconductor microcavity (typically of the size of ), which can reach very high‐quality factors nowadays ( ). Because of the strong ‐axis confinement, the lowest energy eigenstates of a QD in a semiconductor with zincblende crystal structure (e.g., GaAs or InAs) consist of conduction‐band states and valence‐band states. The QDs are doped such that each QD has a full valence band and a single conduction‐band electron: it is assumed that a uniform magnetic field along the ‐direction ( ) is applied, so the qubit is defined by the conduction‐band states |↓〉 and |m _x  = 1/2〉 = |↑〉.

Single‐qubit operations In this scheme, single‐qubit operations are carried out by applying two lasers, polarized along the and directions, that exactly satisfy the Raman‐resonance condition between |↓〉 and |↑〉. The laser fields are turned on for a short time duration that satisfies a ‐pulse condition, where is any real number. The process can be best understood as a Raman ‐pulse for the hole in the conduction‐band state. The laser field polarizations should have nonparallel components in order to create a nonzero Raman coupling.

Two‐qubit operations The two‐qubit operations are mediated by virtual photons that are emitted to and reabsorbed from the microcavity field. It is assumed that the ‐polarized cavity‐mode with energy ( ) and a ‐polarized laser field establish the Raman transition between the two conduction‐band states, in close analogy with the atomic cavity‐quantum electrodynamics (QED) schemes. For a single QD, the Hamiltonian is brought into the form , with

25.37

where , annihilates an electron with spin , along the ‐direction in the conduction band and annihilates an electron with spin along the ‐direction in the valence band. The light–matter interaction has the form

25.38

where is the dipole interaction strength, and the electron creation and annihilation operators in the conduction and valence band, and , denote the cavity and laser mode photon creation and annihilation operators. All the photon and electronic degrees of freedom, except the electron spins in the conduction band, can be eliminated in the case of two quasi‐resonant QDs by a series of formal manipulations ( 9, 16), with the resulting two‐spin Hamiltonian

25.39

where , , and . We have already discussed the implementation of the CPF and the CNOT or quantum XOR gates between two spins and from a transversal (XY) spin coupling of the form 25.39 in Section 25.2.2.1. The interaction Hamiltonian describes the coupling of the QD spins via the following virtual process. One of the QDs emits a virtual photon into the cavity while absorbing a laser photon. The cavity photon is then reabsorbed by the other dot while a laser photon is emitted. Due to the spin splitting in the dot spectrum, this process is spin sensitive and leads to the spin–spin coupling between the QDs.

Measurement Measurement of an individual QD spin in the cavity‐QED scheme can be achieved by applying a laser field to the QD to be measured, in order to realize exact two‐photon resonance with the cavity mode. If the QD spin is in state |↓〉, there is no Raman coupling and no photons will be detected. If on the other hand, the spin state is |↑〉, the electron will exchange energy with the cavity mode and eventually a single photon will be emitted from the cavity. A single photon detection capability is thus sufficient for detecting a single spin.

25.3.3.2 Superconducting Microwave Resonators

Instead of using the electric dipole of spin‐selective interband transitions in the optical frequency range to couple spin qubits, one can alternatively use intraband transitions in the microwave regime related to the dipole of electron motion within one QD or between coupled QDs (34,35). For multispin qubits, the motion of an electron, for example, from the (1,1) to the (0,2) charge state in a double QD, can be coupled to the spin state (singlet or triplet) by the Pauli exclusion principle. In combination with a magnetic field gradient provided, for example, by the resident nuclear spins or a nearby micromagnet, this allows for the coupling of singlet–triplet qubits to a SC microwave cavity (36). In a similar way, three‐spin qubits in triple QDs can be coupled to SC microwave cavities (37,38). For more details, we refer the interested reader to Ref. (21).

25.3.4.1 Phonons and the Spin–Orbit Coupling

The interaction of a confined electron spin with lattice phonons via the spin–orbit interaction can lead to transitions between different discrete energy levels (or Zeeman sublevels) in GaAs QDs that can cause spin flips and therefore spin decoherence (see ( 8, 9) and references therein). Various mechanisms are known, originating from the spin–orbit coupling, which lead to such spin‐flip processes. The most relevant mechanisms in 2D have to do with the broken inversion symmetry, either in the elementary crystal cell or at the heterointerface. The spin–orbit Hamiltonian for the electron in such a structure is given by Eq. 25.16. The spin relaxation rate can be evaluated in leading perturbation order in this coupling, with and without a magnetic field. The spin–orbit coupling mixes the spin‐up and spin‐down states of the electron and leads to a nonvanishing matrix element of the phonon‐assisted transition between two states with opposite spins. However, the spin relaxation of electrons localized in a QD differs strongly from that of delocalized electrons. It turns out that in QDs (in contrast to extended 2D states), the contributions to the spin‐flip rate proportional to are absent. This reduces the spin‐flip rates of electrons confined to dots to a large extent. The finite Zeeman splitting in the energy spectrum also leads to contributions ,

25.40

where is the orbital energy‐level splitting in the QD and the inelastic rate without spin flip for the transition between the neighboring orbital levels.

Spin‐flip transitions between Zeeman sublevels occur with a rate that is proportional to the fifth power of the Zeeman splitting,

25.41

The dimensionless constant characterizes the strength of the effective spin–piezo‐phonon coupling in the heterostructure and ranges from to depending on . As an example, for and at a magnetic field . It was found that under realistic and quite general conditions, a symmetry argument leads to the conclusion that the spin decoherence time has only transverse contributions (in leading order), in other words, for spin–orbit (phonon) related processes (8).

25.3.4.2 Nuclear Spins

The hyperfine interaction between an electron spin and the spins of the surrounding atomic nuclei is another source of electron spin decoherence. A rough estimate of the strength of this effect based on perturbation theory (29) suggests that the rate of such processes can be suppressed by either polarizing the nuclear spins or applying an external magnetic field. The suppression factor is , where is the maximal magnitude of the effective nuclear field (Overhauser field), the number of nuclear spins in the vicinity of the electron, and the hyperfine coupling constant. In GaAs, the nuclear spin of both Ga and As is . The field denotes either the external field, or, in the absence of an external field, the Overhauser field due to the nuclear spin polarization , which can be obtained, for example, by optical pumping or by spin‐polarized currents at the edge of a 2DEG. In the latter case, the suppression of the spin‐flip rate becomes .

A detailed calculation (see (8) for a review) shows that the electron spin decoherence time is shorter than the nuclear spin relaxation time determined by the dipole–dipole interaction between the nuclei, and therefore the problem can be considered in the absence of the nuclear dipole–dipole interaction. Since the hyperfine interaction depends on the position via a factor , where is the electron wavefunction, the value of the hyperfine interaction varies spatially. It turns out that this is the relevant cause of decoherence. The analysis is complicated by the fact that in a weak external Zeeman field (smaller than a typical fluctuating Overhauser field seen by the electron, G in a GaAs QD), the perturbative treatment of the electron spin decoherence breaks down, and the decay of the spin precession amplitude is not exponential in time, but described by either a power law, (for finite Zeeman fields), or an inverse logarithm, (for vanishing fields).

The decoherence rate is thus roughly given by , where is the hyperfine interaction constant and the number of nuclei within the dot, with typically . This time is of the order of several microseconds. However, it needs to be stressed that there is no simple exponential decay which, strictly speaking, means that decoherence cannot simply be characterized by the decay times and in this case. The case of a fully polarized nuclear spin state was solved exactly (8). The amplitude of the precession, which is approached after the decay, is of order one, while the decaying part is , in agreement with the earlier results (29). A large difference between the values of (decoherence time for a single dot) and (dephasing time for an ensemble of dots), that is, , is found and indicates that it is desirable to have direct experimental access to single‐spin decoherence times. We note that the spin coherence time in Si QDs reaches a microsecond in natural Si and exceeds 100 microseconds in isotopically purified ²⁸Si.

25.3.4.3 Charge Noise in Multispin Qubits

The exchange interaction, the spin–orbit coupling, and the Zeeman interaction in an inhomogeneous magnetic field provide coupling mechanisms between the spin and the charge (motional) degrees of freedom of electronic spin qubits. While such spin–charge interactions allow for fast electric control of the spin qubit and the coupling of the spin qubit to the electric field of electromagnetic cavities, they also expose the spin qubit to electrical noise from charge fluctuations in the material, phonons in the surrounding crystal, and electrical noise in high‐bandwidth control lines, leading to the voltage gates that control the QD potential. In this way, a spin qubit can suffer decoherence caused by electrical (charge) noise. For spin qubits controlled with short pulses of the exchange interaction that is turned off in the idle state, the charge noise only reaches the qubit during gate operations. However, those multispin qubits with an always‐on exchange interaction, that is, the hybrid qubit, the RX qubit, and the always‐on exchange qubit, are constantly exposed to charge noise ( 38,41). Therefore, it is favorable to operate such multispin qubits at sweet spots where the influence of charge noise vanishes in lowest order ( 21, 38, 41).

25.4.2.1 Decoherence

We will mostly be concerned with questions related to SC qubit decoherence here. Consider a situation where the qubit is initially prepared in state , that is, in the left well in Figure 25.9. As it evolves under the influence of the Hamiltonian equation 25.42, it will undergo free Larmor oscillations with frequency . Ideally, the probability for finding the qubit in state again after time would be a cosine function , which is plotted as a thin dotted line in Figure 25.9. Such a Larmor precession experiment (also known as Ramsey fringe Experiment) determines to what extent the qubit is quantum phase coherent. Decoherence is a process in which the amplitude of the oscillations decays over time, as shown by the thick solid line in Figure 25.9. This decay is often (but not always) exponential with a characteristic decoherence time .

All types of SC qubits suffer from decoherence that is caused by a number of sources. Decoherence in charge qubits has been investigated using the spin‐boson model (11). A systematic theory of decoherence of a qubit from such dissipative elements, based on the network graph analysis (43) of the underlying SC circuit, was developed for SC flux qubits (44) and applied to study the effect of asymmetries in a persistent‐current qubit (45). The circuit theory for SC qubits will be discussed further in Section 25.4.3 .

25.4.2.2 Visibility

In Figure 25.9, we also show another type of imperfection that typically affects SC qubits: limited visibility . This means that the maximum range of the readout probability of the qubit being in state is smaller than one. The probability of measuring the qubit in state right after preparation in this state is less than 1. In the case of a symmetric reduction of the visibility, the relation is .

25.4.2.3 Leakage

One mechanism (among other mechanisms) leading to a reduced visibility is leakage. Since the SC phase is a continuous variable as, for example, the position of a particle, SC qubits (two‐level systems) have to be obtained by truncation of an infinite‐dimensional Hilbert space. This truncation is only approximate for various reasons: (i) because it may not be possible to prepare the initial state with perfect fidelity in the lowest two states, (ii) because of erroneous transitions to higher levels (leakage effects) due to imperfect gate operations on the system, and (iii) because of erroneous transitions to higher levels due to the unavoidable interaction of the system with the environment. Apparent leakage effects may occur if the readout process is not accurate. Leakage effects due to the nonadiabaticity of externally applied fields were studied in (46). Recent work (47) shows that leakage in microwave‐driven Josephson phase qubits leading to a reduced visibility can occur, even if the microwave source is pulsed slowly.

25.4.3.1 The Hamiltonian

Kirchhoff's laws and the CVRs combined are sufficient to write down the Hamiltonian of an SC circuit. In the absence of dissipative elements (impedances ), the Hamiltonian is

25.46

25.47

where are the charges conjugate to the fluxes and is the capacitance matrix of the circuit. The matrices , , and are obtained from the inductance and loop matrices and (44). The theory is quantized using the commutator relation

25.48

The system including dissipation can be described using the Caldeira–Leggett model,

25.49

25.50

25.51

where is the quantized Hamiltonian equation 25.46 and the Hamiltonian describing a bath of harmonic oscillators with (fictitious) position and momentum operators and with , masses , and oscillator frequencies . Finally, describes the coupling between the system and bath degrees of freedom, and , where is a coupling parameter and are obtained from the inductance and loop matrices and (44).

The time evolution of the qubit (dissipation‐free SC circuit) and oscillator bath (circuit impedances) is determined by the Liouville equation for the density matrix of the combined system. The state of the SC qubit alone can be obtained by taking the partial trace over the harmonic oscillator bath to find the reduced density matrix, . The time evolution for is the master equation, which in general is a complicated linear integro‐differential equation (48). In the Born–Markov approximation, the master equation for can be written in the relatively simple form of the Redfield equations,

25.52

where are the matrix elements of in the eigenbasis of (eigenenergies ), and , and with the Redfield tensor,

25.53

In the two‐dimensional qubit subspace, the Bloch vector can be introduced where are the Pauli matrices, and the Redfield equation 25.52 takes the form of the Bloch equation , with , where in the secular approximation, the relaxation matrix is diagonal, . The relaxation and decoherence times and are then given by

25.54

25.55

25.56

In the semiclassical approximation, and can be related to the parameters and in the Hamiltonian equation 25.42,

25.57

25.58

25.4.3.2 The Delft Qubit

A very successful qubit design is the Delft qubit (42), which is depicted in Figure 25.8, and which will be discussed in this Section. A schematic drawing of the SC circuit for the Delft qubit is shown in Figure 25.11. This design is intended to be immune to current fluctuations in the current bias due to its symmetry properties; at zero dc bias, , and independent of the applied magnetic field, a small fluctuating current caused by the finite impedance of the external control circuit (the current source) is divided equally into the two arms of the SQUID loop, and no net current flows through the three‐junction qubit line. Hence, in the ideal circuit (Figure 25.11) the qubit is protected from decoherence due to current fluctuations in the bias current line. However, asymmetries in the SQUID loop may spoil the protection of the qubit from decoherence. In the case of an inductively coupled SQUID (4,49,50), neither a small geometrical asymmetry (imbalance of self‐ and mutual inductances in the SQUID loop) nor the junction (critical current) asymmetry of typically a few percent would suffice to cause a relevant amount of decoherence at zero bias current (44). What turns out to be important here is that the circuit (Figure 25.8) contains another asymmetry, caused by its double‐layer structure, being an artifact of the fabrication method used to produce SC circuits with aluminum/aluminum oxide Josephson junctions, the so‐called shadow evaporation technique. Junctions produced with this technique will always connect the top layer with the bottom layer, see Figure 25.12.

Thus, while circuits such as that shown in Figure 25.11 can be produced with this technique, strictly speaking, loops will always contain an even number of junctions. In order to analyze the implications of the double‐layer structure for the circuit in Figure 25.11, the circuit is drawn again in Figure 25.13a with separate upper and lower layers. Note that each piece of the upper layer is connected with the underlying piece of the lower layer via an “unintentional” Josephson junction.

These extra junctions typically have large areas and therefore large critical currents; thus, their Josephson energy can often be neglected. In order to study the lowest order effect of the Double‐layer structure, one can neglect all unintentional junctions in this sense and arrive at the circuit Figure 25.13b. It should be emphasized that the resulting circuit is distinct from the “ideal” circuit Figure 25.11 which does not reflect the double‐layer structure. In the real circuit, Figure 25.13b, the symmetry between the two arms of the dc SQUID is broken, and thus it can be expected that bias current fluctuations cause decoherence of the qubit at zero dc bias current, .

Starting from the circuit graph of the Delft qubit, the circuit theory can be used to find the Hamiltonian of the circuit, which can subsequently be analyzed numerically. The double‐well minima and were found for a range of bias currents and applied external flux. The states localized at and are encoding the logical and states of the qubit. Two special lines in the plane spanned by the bias currents and applied external flux can now be determined, see Figure 25.14. (i) The line on which a symmetric double well is predicted, . On this line, the dephasing time diverges. (ii) The line on which , where is the vector joining the two minima of the potential. On this line, the environment is decoupled from the system, and both the relaxation and the decoherence times diverge, . The curve agrees qualitatively with the experimentally measured symmetry line (51), but it underestimates the magnitude of the variation in flux as a function of . The point where the symmetric and the decoupling lines intersect coincides with the maximum of the symmetric line, as can be understood from the following argument. Taking the total derivative with respect to of the relation on the symmetric line, and using that are extremal points of , we obtain for some constant vector . Therefore, (decoupling line) and implies .

The relaxation and decoherence times and on the symmetric line have been evaluated and are plotted (Figure 25.14, right) where , and therefore, . The divergence in on the symmetric line is cut off by higher order effects, whereas the divergence of on the decoupling line is cut off by residual impedances, for example, due to the junctions' quasi‐particle resistance (45). A peak in the relaxation and decoherence times where predicted from theory can be observed experimentally (51).