Quantum Information Processing with Superconducting Circuits: a Review
Abstract
During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for nearfuture useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of Quantum Supremacy with fifty qubits is anticipated in just a few years. Quantum Supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classicalquantum computing systems are already emerging that can be used for software development and experimentation, even via web interfaces.
Therefore, the time is ripe for describing some of the recent development of superconducting devices, systems and applications. As such, the discussion of superconducting qubits and circuits is limited to devices that are proven useful for current or near future applications. Consequently, the centre of interest is the practical applications of QIP, such as computation and simulation in Physics and Chemistry.
Keywords: superconducting circuits, microwave resonators, Josephson junctions, qubits, quantum computing, simulation, quantum control, quantum error correction, superposition, entanglement
Contents
 1 Introduction
 2 Easy and hard problems
 3 Superconducting circuits and systems
 4 Transmon quantum circuits
 5 Hybrid circuits and systems
 6 Quantum gates
 7 Quantum state preparation and characterisation
 8 Quantum state protection

9 Quantum simulation of manybody systems
 9.1 Basics of quantum simulation
 9.2 Trotterisation
 9.3 Phase estimation
 9.4 Digital quantum simulation of the quantum Rabi model
 9.5 Digital quantum simulation of spin models
 9.6 Digital quantum simulation of fermionic models
 9.7 Analogue/adiabatic quantum simulation
 9.8 Digitalanalogue quantum simulation
 10 Toward quantum chemistry simulation
 11 Adiabatic quantum optimisation
 12 Perspectives
1 Introduction
Quantum Computing is the art of controlling the time evolution of highly complex, entangled quantum states in physical hardware registers for computation and simulation. Quantum Supremacy is a recent term for an old ambition  to prove and demonstrate that quantum computers can outperform conventional classical computers [1].
Since the 1980’s, quantum computer science has been way ahead of experiment, driving the development of quantum information processing (QIP) at abstract and formal levels. This situation may now be changing, due to recent experimental advances to scale up and operate highly coherent and operational qubit platforms. In particular, one can expect superconducting quantum hardware systems with 50 qubits or more during the next few years.
The nearterm goal is to operate a physical quantum device that a classical computer cannot simulate [2], therefore demonstrating Quantum Supremacy. For QIP to be of interest one often requests ”killer applications”, outperforming classical supercomputers on realworld applications like factorisation and code breaking [3]. However, this is not a realistic way to look upon the power of QIP. During the last seventy years, classical information processing has progressed via continuous development and improvement of more or less efficient algorithms to solve specific tasks, in tune with the development of increasingly powerful hardware. The same will certainly apply also to QIP, the really useful applications arriving along the way.
There is, in fact, already a clear shortterm QIP perspective, recognising the importance of quantum technologies (QT) and quantum engineering for driving the present and nearfuture development [4]. This is necessary for developing large scale devices to achieve the longterm goals of useful quantum computing. To this end, investigations of quantum device physics have been essential for the present efforts to scale up of multiqubit platforms. The work on improving coherence has demonstrated that qubits are extremely sensitive to a variety of noise sources. The requirement of making measurements without destroying the coherence has led to the development of a range of quantum limited superconducting amplifiers. As a result, advanced quantum sensors and quantum measurement may give rise to a new quantum technology: ”a qubit at the tip of a scanning probe”, greatly enhancing the sensitivity of magnetic measurements [5, 6, 7].
The potential of superconducting circuits for QIP has been recognised for more than twenty years [8, 9, 10]. The first experimental realisation of the simplest qubit, a Josephsonjunction (JJ) based Cooperpair Box (CPB) in the charge regime, was demonstrated in 1999 [11]. Originally, the coherence time was very short, just a few nanoseconds, but it took only another few years to demonstrate a number of useful qubit concepts: the flux qubit [12, 13, 14], the quantronium CPB [15], and the phase qubit [16]. The next important step was to embed qubits in a superconducting microwave resonator, introducing circuit quantum electrodynamics (cQED)[17, 18, 19, 20, 21, 22]. The subsequent experiments using a 2D superconducting coplanar microwave resonator [23, 24] demonstrated groundbreaking progress:

Microwave qubit control, strong qubitresonator coupling and dispersive readout [23];

Coupling of CPB qubits and swapping excitations, in practice implementing a universal gate [24].
A basis for potentially scalable multiqubit systems with useful long coherence times  the transmon version of the CPB  was published in 2007 [25]. Moreover, in 2011 the invention of a transmon embedded in a 3Dcavity increased coherence times toward 100 s [26]. At present there is intense development of both 2D and 3D multiqubit circuits with longlived qubits and resonators, capable of performing a large number of highfidelity quantum gates with control and readout operations.
The purpose of this review is to provide a snapshot of current progress, and to outline some expectations for the future. We will focus on hardware and protocols actually implemented on current superconducting devices, with discussion of the most promising development to scale up superconducting circuits and systems. In fact, superconducting quantum circuits are now being scaled up experimentally to systems with several tens of qubits, to address real issues of quantum computing and simulation [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].
The aim is here to present a selfcontained discussion for a broad QIP readership. To this end, time evolution and the construction and implementation of 1q and 2q gates in superconducting devices are treated in considerable detail to make recent experimental work more easily accessible. On the other hand, the more general discussion of theory, as well as much of the experimental work, only touches the surface and is covered by references to recent work. Reviews and analyses of the QIP field are provided by [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. For a comprehensive treatise on QIP we refer to Nielsen and Chuang [54]. Extensive technical discussions of a broad range of superconducting circuits and qubits can be found in [55, 56, 57, 58, 59, 60].
The present review also tries to look beyond the experimental state of the art, to anticipate what will be coming up in the near future in the way of applications. There is so much theoretical experience that waits to be implemented on superconducting platforms. The ambition is to outline opportunities for addressing realworld problems in Physics, Chemistry and Materials Science.
2 Easy and hard problems
Why are quantum computers and quantum simulators of such great interest? Quantum computers are certainly able to solve some problems much faster than classical computers. However, this does not say much about solving computational problems that are hard for classical computers. Hard problems are not only a question of whether they take long time  the question is whether they can be solved at all with finite resources.
The map of computational complexity in Fig. 1 classifies the hardness of computational (decision) problems. Quantum computation belongs to class BQP (boundederror quantum polynomial time). Figure 1 shows that the BQP only encompasses a rather limited space, basically not solving really hard problems. One may then ask what is the relation between problems of practical interest and really hard mathematical problems  what is the usefulness of quantum computing, and which problems are hard even for quantum computers?
2.1 Computational complexity
Computational complexity [1, 62, 63, 64] is defined in terms of different kinds of Turing machines (TM) providing digital models of computing. A universal TM (UTM) can simulate any other TM (including quantum computers) and defines what is computable in principle, without caring about time and memory. Problems that can be solved by a deterministic TM (DTM) in polynomial time belong to class P (Fig. 1), and are considered to be ”easy”, or at least tractable. A DTM is a model for ordinary classical computers  a finite state machine (FSM) reading and writing from a finite tape.
A probabilistic TM (PTM) makes random choices of the state of the FSM upon reading from the tape, and traverses all the states in a random sequence. This defines the class BPP (boundederror probabilistic polynomial time). A PTM may be more powerful then a DTM since it avoids getting stuck away from the solution. Nevertheless, a PTM can be simulated by a DTM with only polynomial overhead, so the relation BPP=P is believed to be true.
A quantum TM (QTM) is a model for a quantum computer with a quantum processor and quantum tape (quantum memory). Problems that can be solved by a QTM in polynomial time belong to class BQP (Fig. 1). There, outside P (P BQP), we find problems like Shor’s algorithm [3] where a QTM provides exponential speedup. Nevertheless, Fig. 1 shows that BQP is a limited region of the complexity map, not including a large part of the NPclass containing many hard problems for a classical computer. It is unknown whether these are hard for a quantum computer. The class NP (nondeterministic polynomial) is defined by a nondeterministic TM (NTM), able to provide an answer that can be verified by a DTM in polynomial time. The NTM is not a real computer but rather works as an oracle, providing an answer. A subclass of NP is the MA (MerlinArthur) class where the allmighty Merlin provides the answer which the classical Arthur can verify in polynomial time. Some of these problems are beyond a quantum computer to calculate, but it might be used to verify solutions in polynomial time. This is the large quantum MerlinArthur (QMA) complexity class shown in Fig. 1. When not even a quantum computer can verify a solution in polynomial time, then that problem belongs to the NPhard complexity class, where even a quantum computer is of no use.
2.2 Hard problems
Can a physical processes exist while being too hard to compute? Or does Nature actually not solve really hard instances of hard problems? Perhaps the results of Evolution are based on optimisation and compromises?
There is a longstanding notion that unconventional adaptive analogue computers can provide solutions to NPhard problems that take exponential resources (time and/or memory) for classical digital machines to solve [65, 66, 67, 68, 69, 70, 71]. The key question therefore is: can unconventional computing provide solutions to NPcomplete problems? The answer is in principle given by the Strong Church Thesis: Any finite analogue computer can be simulated efficiently by a digital computer. This is due to the time required by the digital computer to simulate the analogue computer being This is due to the time required by the digital computer to simulate the analogue computer being bounded by a polynomial function of the resources used by the analogue machine [72]. This suggests that physical systems (both digital and analogue) cannot provide solutions to NPcomplete problems. NPcompleteness is a worst case analysis, where at least one case requires exponential, rather than polynomial, resources in the form of time or memory.
Given the idea that ”Nature is physical and does not solve NPhard problems” [61, 73, 74, 75], where does this place quantum computing? In principle, in a better position than classical computing, to compute the properties of physical quantum systems. Tractable problems for quantum computers (BQP) are in principle hard for classical computers (P). In 1982 Feynman [76] introduced the concept of simulating one quantum system by another, emulating quantum physics by tailored quantum systems describing model quantum Hamiltonians (analogue quantum computers). The subsequent development went mostly in the direction of gate circuit models [77], but two decades later the analogue/adiabatic approach was formally established as an equivalent universal approach [78, 79, 80, 81, 82]. Nevertheless, also for quantum computers the class of tractable problems (BQP) is limited. There are many problems described by quantum Hamiltonians that are hard for quantum computers, residing in QMA, or worse (QMAhard, or NPhard) [83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94]. Although, the manybody problem is tractable for quantum spin chains [95, 96]. When quantum mechanics is not involved, e.g. in combinatorial problems, quantum computers may not have any advantage [61].
2.3 Quantum speedup
Quantum speedup is by definition connected with nonclassical correlations [97, 98]. Entanglement is a fundamental manifestation of quantum superposition and nonclassical correlations for pure states [99]. An elementary example of classical behaviour is provided by a tensor product of independent superpositions of two 2level systems, and . The tensor product of (= 2) states contains (= 4) superposed configurations: this is the basis for creating exponentially large superpositions with only a linear amount of physical resources (qubits).
Highly entangled states are created by finite numbers of superpositions. In the present 2qubit case, a maximally entangled state is the 2qubit Bell state, : it is not possible to assign a single state vector to any of the two subsystems, only to the total system. Entanglement allows us to construct maximally entangled superpositions with only a linear amount of physical resources, e.g. a large cat state: , entangling 2level systems. This is what allows us to perform nonclassical tasks and provide speedup [97, 98]. Interestingly, just to characterise the entanglement can be a hard problem for a classical computer, because several entanglement measures are NPhard to compute [100]. There is a large number of measures of entanglement, e.g. concurrence; entropy of entanglement (bipartite); entanglement of formation; negativity; quantum discord [98, 100, 101, 102, 103, 104, 105, 106]. Quantum discord is defined as the difference between two classically equivalent measures of information [107], and indicates the presence of correlations due to noncommutativity of quantum operators. For pure states it equals the entropy of entanglement [102]. Quantum discord determines the interferometric power of quantum states [104]. It provides a fundamental concept for computation with mixed quantum states in open systems, separable and lacking entanglement but still providing useful nonclassical correlations.
Quantum speedup is achieved by definition if a quantum calculation is successful, as discussed by Dewes et al. [108] in the case of Grover search with a transmon 2qubit system. It is then related to the expected known success probabilities of the classical and quantum systems. In general, however, speedup of a computation is an asymptotic scaling property [109]. Nevertheless, in practice there are so many different aspects involving setting up and solving different instances of various classes of problems that the time to solution (TTS) may be the most relevant measure [110].
Polynomial or exponential speedup has not been much discussed in connection with digital QC because the systems are still small (510 qubits), and the limited coherence time does not allow very long calculations. In contrast, defining and detecting quantum speedup is presently a hot issue when assessing the performance of the DWave quantum annealing machines [109, 110, 111, 112].
2.4 Quantum Supremacy
Quantum Supremacy is a recent term for the old ambition of proving that quantum computers can outperform conventional classical computers [1]. A simple implementation of Quantum supremacy is to create a physical quantum device that cannot be simulated by existing classical computers with available memory in any reasonable time. Currently, such a device would be a 50 qubit processor. It could model a large molecule that cannot be simulated by a classical computer. It would be an artificial physical piece of quantum matter that can be characterised by various quantum benchmarking methods in a limited time, but cannot be simulated by classical computers of today. And by scaling up by a small number of qubits it will not be simulatable even by next generation classical computers. A recent example of this given by Boixo et al. [2], discussing how to characterise Quantum Supremacy in nearterm circuits and systems with superconducting devices.
3 Superconducting circuits and systems
The last twenty years have witnessed a dramatic development of coherent nano and microsystems. When the DiVincenzo criteria were first formulated during 19962000 [113, 114] there were essentially no useful solidstate qubit devices around. Certainly there were a number of quantum devices: Josephson junctions (JJ), Cooper pair boxes (CPB), semiconductor quantum dots, implanted spins, etc. However, there was no technology for building coherent systems that could be kept isolated from the environment and controlled at will from the outside. These problems were addressed through a steady technological development during the subsequent ten years, and the most recent development is now resulting in practical approaches toward scalable systems.
3.1 The DiVincenzo criteria (DV1DV7)
The seven DiVincenzo criteria [114] formulate necessary conditions for gatedriven (digital) QIP:
1. Qubits: fabrication of registers with several (many) qubits (DV1).
2. Initialisation: the qubit register must be possible to initialise to a known state (DV2).
3. Universal gate operations: high fidelity single and 2qubit gate operations must be available (DV3).
4. Readout: the state of the qubit register must be possible to read out, typically via readout of individual qubits (DV4).
5. Long coherence times: a large number of single and 2qubit gate operations must be performed within the coherence time of the qubit register, (DV5).
6. Quantum interfaces for qubit interconversion: qubit interfaces must be possible for storage and onchip communication between qubit registers (DV6).
7. Quantum interfaces to flying qubits for optical communication: qubitphoton interfaces must be available for longdistance transfer of entanglement and quantum information (DV7).
3.2 Josephson quantum circuits
The recent systematic development of reliable transmonbased JJcQED circuits is now forming a basis for serious upscaling to 50 qubits in the near future, to develop system control, benchmarking, error correction and quantum simulation schemes.
The qubits are based on the superconducting nonlinear oscillator circuit shown in Fig. 2 (for an indepth discussion, see Girvin [57]). The Hamiltonian of the harmonic oscillator LC circuit alone is given by
(1) 
where is the induced charge on the capacitor measured in units of 2e (Cooper pair), and is the phase difference over the inductor. The charge and phase operators do not commute, , which means that their expectation values cannot be measured simultaneously. (charging energy of one 2e Cooper pair), , and the distance between the energy levels of the harmonic oscillator is given by .
For a superconducting highQ oscillator the energy levels are narrow and equidistant. However, in order to serve as a qubit, the oscillator must be anharmonic so that a specific pair of levels can be addressed. Adding the Josephson junction (JJ), the Hamiltonian of the LCJ circuit becomes
(2) 
where is the voltageinduced charge on the capacitor C (qubit island), and is the fluxinduced phase across the JJ. The Josephson energy is given by in terms of the critical current of the junction [55]. Typically, the JJ is of SIS type (superconductorinsulatorsuperconductor) with fixed critical current.
In order to introduce the Josephson nonlinear inductance, one starts from the fundamental Josephson relation
(3) 
Combined with Lenz’ law:
(4) 
one finds that
(5) 
Defining , one finally obtains the Josephson inductance :
(6) 
This defines the Josephson inductance of the isolated JJ circuit element in Fig. 2, and allows us to express the Josephson energy as .
In order to describe the energylevel structure of the quantum LCJ circuit in Fig. 2 one introduces to get a Schrödinger equation for the circuit wave function in the phase variable :
(7) 
(8) 
(1) =0 () : becomes a pure cosine periodic potential, and the wave function has the form , where is a Mathieu function. The energy levels form bands in the ”momentum” direction [22, 25]. The dispersion of these band depends on the ratio , as shown in Fig. 3. Of special interest is that a large capacitance results in flat lowlying bands, making the circuit insensitive to charge fluctuations (as well as to charge control via a DC gate voltage) [25].
(2) : is no longer periodic, but described by a parabola modulated by sinusoidal function. The shape of the potential toward the bottom of the parabola, and the associated qubit level structure, depend on the ratio and on the external flux . This makes it possible to design a wide variety of qubits by tuning the circuit parameters in Fig. 2.
3.3 Qubits (DV1)
We will now briefly discuss the qubit families listed in Table 1 and shown in Fig. 4. In reality, all ”qubits” are multilevel systems. However, since they are most often treated as quantum bits (binary logic), we will refer to them as qubits even if additional levels are used for gate operations. The qubits can be defined in terms of different values of the circuit parameters through the and ratios, characterising a number of charge and flux types of devices.
Z  
fF  fF  pH  pH  
1. Phase qubit [16]  0  6000  3300  16  0.005  10  1.5 
2. Phase qubit [115]  800  0  720  80  0.11  10  15 
3. rfSQUID [12]  0  40  238  101  0.43  2000  48 
4. Flux qubit [125]  0  3  1200  600  0.5  10  450 
5. Fluxonium [137]  0.15  0  3300  150  0.045  1  1400 
6. Cshunt [140]  50  0  15000  4500  0.3  25  480 
7. Charge qubit [11]  0.68  0  808  0  0.018  10  
8. Quantronium [15]  2.8  0  1.1 10  0  1.27  1300  
9. Transmon [25, 26]  1540  0  10  0  1050  250  
10. Xmon [149]  100  0  10  0  2228  500  
11. Gatemon [153]  100  0  10  0  1732  500 
Charging energy of one Cooper pair (2e):
Inductive energy:
Josephson energy:
Resistance ”quantum”:
Impedance:
3.3.1 Phase qubit
The phase qubit is formed by the two lowest levels in the potential wells formed by a currentbiased Josephson junction. In practice, current bias is achieved by fluxbiasing an rfSQUID ( knob) (Fig. 2), placing the qubit in an anharmonic potential well on the slope of the parabola.
The original phase qubit [16] was built on a largearea junction with large selfcapacitance (Table 1). However, the large area of the junction oxide gave rise to many defects, trapping twolevel fluctuators (TLS) that severely limited the coherence time. An improved phase qubit [115] was created by separating the device into a smallarea (low ) JJ with the same critical current (and thus the same , ), and a large shunt capacitance with a dielectric with much fewer defects. This phase qubit was the first one to be used for advanced and groundbreaking QIP applications with up to four qubits [116, 117, 118, 119, 120, 121]. However, the coherence time has stayed rather short (); therefore, phasequbit technology cannot be scaled up at the present time. See [22, 55] for detailed discussions.
3.3.2 rfSQUID flux qubit
The rfSQUID flux qubit [12, 122] is a persistentcurrent qubit obtained by setting the biasing flux to so that the Josephson part creates two potential wells separated by a barrier at the bottom of the parabola. This defines two lowlying ”bondingantibonding” qubit levels describing superpositions of left and rightrotating supercurrents: . Since the inductive SQUID loop is large, this flux qubit is sensitive to flux noise, and the relaxation and coherence times are quite short, [122], probably due to twolevel fluctuators in the Nb/AlOx/Nb trilayer junction [122]. The DWave Systems’ flux qubit is of this type [123, 124].
3.3.3 ThreeJJ flux qubit
The threeJJ flux qubit [13, 14, 125] consists of an rfSQUID where the inductor has been replaced by two JJs to provide large inductance with a small SQUID ring. Since the added JJs also create an oscillating cosine potential, with the right parameters there appears a periodic potential with a double well at the bottom of each major well, defining two lowlying ”bondingantibonding” qubit levels. Tuning the flux bias with the knob makes it possible to vary the relative energies of the wells. Since the 3JJ potential is periodic, it is associated with a band structure.
The threeJJ flux qubit has always been a major candidate for scaledup multiqubit systems, but the coherence time has not improved much, which has so far limited applications to cases making use of the SQUID properties and strong flux coupling [126, 127, 128] for applications to microwave technology [129, 130], analog computing [131], and metamaterials [132, 133, 134]. A recent experiment has demonstrated somewhat longer coherence time of a flux qubit in a 3D cavity [135]. Also, see the Cshunt flux qubit below.
3.3.4 Fluxonium qubit
The fluxonium [136, 137] consists of a small JJ shunted by a very large inductance provided by a long array of large JJs. The resulting effective capacitance is very small (see Table 1). This looks similar to the 3JJ flux qubit in the sense that the two large JJs are replaced by a large JJ array. Approximately, the large array creates a wider parabola accommodating several potential wells. An important thing is that the capacitance C is so small that the there are practically no charge fluctuations (similar to the CPB charge qubit). The relaxation time at 1/2 flux quantum bias is due to suppression of coupling to quasiparticles [137].
3.3.5 Cshunt flux qubit
The Cshunt flux qubit [138, 139] is usually viewed as a 3JJ flux qubit shunted by a large capacitance. Viewed from another angle, it can be also be viewed as a transmon shunted by the effective large inductance of the two large flux qubits of the 3JJ flux qubit. The effect is to flatten the bottom of the wells of the transmon cosine potential, making them quartic rather than quadratic. There is then no longer any doublewell structure like in the flux qubit, but still strong anharmonicity (in contrast to the transmon)
Experimentally, presently the Cshunt flux qubit shows great promise [140], with broad frequency tunability, strong anharmonicity, high reproducibility, and coherence times in excess of 40 s at its fluxinsensitive point.
3.3.6 2D Transmon qubit
The transmon [25] is a development of the CPB toward a circuit with low sensitivity to charge noise, and therefore much longer coherence times. This is achieved by radically flattening the bands in the charge direction by increasing the ratio (Fig. 3d). It should be noted that the transmon is really a flatband multilevel system (qudit), and the higher levels are often used for implemetation of 2qubit gates. Since the influence of the charge offset will vanish (the situation in Fig. 3d), it follows that the energy levels can no longer be tuned statically by the charge gate  it can only be used for microwave excitation to drive transitions between energy levels. The driving is differential  the transmon is floating (not grounded). Tuning of the frequency of the transmon by varying the Josephson energy can be achieved by replacing the JJ by a 2JJ SQUID (which then also increases the sensitivity to flux noise).
3.3.7 3D Transmon qubit
In the 3D transmon [26], the JJ qubit is coupled to the 3D cavity through a broadband planar dipole antenna. Experiments with 3D devices and architectures are presently demonstrating important progress along two different lines: (i) like in 2D, a digital qubit approach with 1q and 2q gate operations controlled by microwave driving [141]; (ii) a continuousvariable approach where the 3D cavities carry the information in multiphoton ”catstates”, and the transmon qubits mainly serve for creating and controlling the states of the cavities [142, 143, 144, 145, 146, 147, 148].
3.3.8 Xmon qubit
The Xmon [32, 35, 36, 37, 149, 150] is a transmontype qubit developed for architectures with 2D arrays of nearestneighbour capacitively coupled qubits. The large shunt capacitance (see Table 1) has the shape of a cross and is grounded via the JJSQUID, allowing tunability of the qubit frequency.
The Xmon is established as a component for a major scalable platform. Circuits and systems with up to 9 Xmon qubits [35, 37] are presently being investigated with applications to a wide range of QIP problems.
A variation is the gmon with direct tunable coupling between qubits [151].
3.3.9 Gatemon qubit
The gatemon [152, 153] is a new type of transmonlike device, a semiconductor nanowirebased superconducting qubit. The gatemon is of weaklink SNS type (superconductornormalmetalsuperconductor), and the Josephson energy is controlled by an electrostatic gate that depletes carriers in a semiconducting weak link region, i.e. controls the critical current like in a superconducting transistor. There is strong coupling to an onchip microwave cavity, and coherent qubit control via gate voltage pulses. Experiments with a twoqubit gatemon circuit has demonstrated coherent capacitive coupling, swap operations and a twoqubit controlledphase gate [153].
3.3.10 Andreev level qubit
The Andreev level qubit (ALQ) [154, 155] is a spindegenerate, singlechannel, SNStype Josephson junction in an rfSQUID loop. The ALQ can be strongly coupled to a coplanar resonator [156]. Adding coupling to a spin degree of freedom in the junction makes it possible to manipulate the Andreev bound states (ABS) with a magnetic field [157, 158].
Recently there has been significant experimental progress toward detecting and manipulating ABSs in atomic contacts (break junction point contacts) [159, 160] and hybrid semiconductorsuperconductor (SmS) nanostructures [161, 162]. The device is typically an InAs nanowire (NW) between epitaxially grown superconducting Al electrodes (S). The resulting SNWS Josephson junction [162] is in fact a singlechannel version of the gatemon [153]. The ALQ is potentially longlived, but so far the coherence times are short  the ALQ remains a device for fundamental research.
3.3.11 Majorana qubit
The ultimate system for quantum computing might be devices based on topological protection of information. One such system could be Majorana bound states (MBS) in SmS nanostructures that produce ABS at the interface between the normal NW semiconductor (Sm) and the superconductor (S) [163, 164, 165, 166, 167]. By applying an axial magnetic field along the SNW device, one can make the ABSs move to zero energy with increasing magnetic field and form mid gap states [161, 167]. If the states remain at zero energy in a long junction, a topological phase forms with MBSs at the endpoints of the nanowire [167]. The first experimental signatures of MBS in superconductorsemiconductor nanowires [164] have been confirmed [161, 167], and extended to superconductoratomic chain platforms [168]. A major issue is how to manipulate topologically protected qubits to allow universal quantum computation [169, 170].
3.4 Initialisation (DV2)
3.5 Universal gate operation (DV3)
Universal high fidelity single and twoqubit operations (Clifford + T gates; see Sect. 6.6) have been achieved for all major types of superconducting qubits. The shortest time needed for basic 1 and 2qubit quantum operation is a few nanoseconds. Entangling gates with 99.4% fidelity have recently been demonstrated experimentally [32]. However, it should be noted that highfidelity gates may require carefully shaped control pulses with typically 1040 s duration.
3.6 Readout (DV4)
There are now wellestablished efficient methods for singleshot readout of individual qubits, typically performed via dispersive readout of a resonator circuit coupled to the qubit. A strong measurement ”collapses” the system to a specific state, and then repeated nondestructive measurements will give the same result. Singleshot measurements require extremely sensitive quantumlimited amplifiers, and it is the recent development of such amplifiers that has made singleshot readout of individual qubits possible [48, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186].
3.7 Coherence times (DV5)
JJqubits are manufactured and therefore sensitive to imperfections. Nevertheless, there has been a remarkable improvement of the coherence times of both qubits and resonators during the last five years [48, 150, 187, 188]. Table 2 indicates the present state of the art.
2D Tmon  3D Tmon  Xmon  Fluxm  Cshunt  Flux  Gatemon  
[25]  [26]  [149]  [137]  [139, 140]  [13]  [153]  
DV1, #q  5 [34, 204]  4 [141]  9 [32, 37]  1  2 [140]  4 [205]  2 
DV2  
DV3  
t  1020  3040  1020  510  510  30  
n  
F  0.999  0.9995  
t  1040  450  530  50  
n  
F  0.99  0.960.98  0.9945  0.91  
DV4  
DV5  
40  100  50  1000  55  20 [135]  5.3  
40  140  20  40  3.7  
40  140  85  9.5 
 The number of qubits () refers to operational circuits with all qubits connected.
 and are gate times for 1q and 2qgates.
 and are the number of 1q and 2qgate operations in the coherence time.
 and are average fidelities of 1q and 2qgates, measured e.g. via randomised benchmarking (Sect. 7.1).
 is the qubit energy relaxation time.
 is the qubit coherence time measured in a Ramsey experiment.
 is the qubit coherence time measured in a spinecho (refocusing) experiment.
 Table entries marked with a hyphen () indicate present lack of data.
 Note that average gate fidelities F and F do not necessarily correspond to thresholds for error correction [207].
 The t gate time for the 3D Tmon refers to a resonatorinduced phase gate.
3.8 Algorithms, protocols and software
A number of central quantum algorithms and protocols have been performed experimentally with multiqubit circuits and platforms built from the main types of superconducting JJbased qubits (see e.g. [36, 37, 108, 120, 189, 190, 191, 192, 193]), demonstrating proofs of principle and allowing several transmontype systems to be scaled up.
In practice, a quantum computer (QC) is always embedded in a classical computer (CC), surrounded by several classical shells of hardware (HW) and software (SW) [31, 194, 195]. Quantum computation and quantum simulation then involve a number of steps:

CC control and readout HW (shaped microwave pulses, bias voltages, bias fluxes).

CC control and readout SW for the HW (”machine language”; optimal control).

CC subroutines implementing gates (gate libraries).

Highlevel CC optimal control of quantum operations.

CC subroutines implementing quantum gate sequences (for benchmarking, QFT, time evolution, etc.).
The only truly quantum part is step (v), explicitly performing quantum gates on quantum HW and quantum states. This is where quantum speedup can be achieved, in principle. Since the quantum gates have to be implemented by classical SW, it is necessary that the needed number of gates to describe a quantum circuit scales polynominally in the size of problem. For the Clifford gates there are efficient (polynomial) representations. However, to describe an arbitrary, universal quantum circuit needs Tgates and may take exponential resources [202].
If the quantum gates are executed in SW on representations of quantum states on a classical machine, then the quantum computer is emulated by the classical machine. Then to execute the gates scales exponentially, which means that a classical computer can only simulate a small quantum system. The present limit is around 50 qubits [2, 202]  beyond that is the realm of Quantum Supremacy [2] .
4 Transmon quantum circuits
The present development of quantum information processing with scalable Josephson Junction circuits and systems goes in the direction of coupling transmontype qubits with quantum oscillators, for operation, readout and memory. In this section we will therefore focus on the transmon, and describe the components in some detail. For an indepth discussion, the reader is referred to the original article by Koch et al. [25].
4.1 Transmon cQED
A generic compact circuit model for the device is shown in Fig. 5a, and a hardware implementation is shown schematically in Fig. 5b.
The transmon circuit in Fig. 5a consists of a number of fundamental components:
 A Cooper pair box (CPB) with one or two Josephson junctions (JJ) sitting in a closed circuit with a large shunting capacitance (”anharmonic oscillator”). The excitation energy of the twolevel systems is
 A resonator circuit (harmonic oscillator) with frequency .
 A capacitance coupling the transmon and the resonator with coupling constant .
 A drive circuit (right) fluxcoupled to the SQUIDtype JJ circuit for tuning the qubit energy.
 A microwave drive circuit (left) capacitively coupled to the CPW for qubit operation.
 There is no explicit readout oscillator included in Fig. 5a, but the bus resonator can serve to illustrate a readout device..
Treating the transmon as an approximate twolevel system with linear coupling to a singlemode oscillator, the transmoncQED Hamiltonian takes the form
(9) 
where is the qubit excitation energy, is the qubitoscillator coupling, and is the oscillator frequency.
Introducing the raising and lowering operators , the qubitresonator coupling term is split into two terms, JaynesCummings (JC) and antiJaynesCummings (AJC):
(10) 
This Hamiltonian describes the canonical quantum Rabi model (QRM) [208, 209, 210]. Equations (9,10) are completely general, applicable to any qubitoscillator system. Only keeping the JaynesCummings (JC) terms corresponds to performing the rotatingwave approximation (RWA).
4.2 Weak, strong, ultrastrong and deepstrong coupling
There are five basic energy scales that determine the qubitoscillator coupling strength: , plus the oscillator decay rate (resonance line width) and the qubit decay rate (transition line width) .
One typically distinguishes between four cases of qubitoscillator coupling:
(i) Weak coupling: ; RWA valid for .
(ii) Strong coupling: ; RWA valid; vacuum Rabi oscillations.
(iii) Ultrastrong coupling (USC): ; RWA breaks down; counterrotating term important.
(iv) Deepstrong coupling (DSC): ; RWA not valid at all; essential; qubitoscillator compound system.
In the two cases of weak and strong coupling, one performs the rotatingwave approximation (RWA) and only keeps the first term, which gives the canonical JaynesCummings model [208, 211, 212],
(11) 
describing dipole coupling of a twolevel system to an oscillator. In the nonresonant case, diagonalising the JaynesCummings Hamiltonian to second order by a unitary transformation gives [21, 25, 212]
(12) 
where is the socalled detuning. The result implies that (i) the qubit transition energy is Stark shifted (renormalized) by the coupling to the oscillator, and (ii) the oscillator energy is shifted by the qubit in different directions depending on the state of the qubit. This condition allows discriminating the two qubit states in dispersive readout measurement [21, 23].
The strong coupling situation [17, 18, 21] was demonstrated experimentally already in 2004 with superconducting CPBcQED [23] by direct physical coupling of the CPB and the 2D resonator. The ultrastrong coupling case [209, 213] is more difficult to achieve by direct statical physical coupling of a transmon qubit and a resonator, and but has recently been achieved experimentally using flux qubit cQED [126, 127, 128]. On the other hand, it is possible to simulate the QRM in the USC and DSC regimes by external timedependent driving of the oscillator in analogue [213, 214] or digital [215, 216, 217] quantum simulation schemes.
4.3 Multiqubit Transmon Hamiltonians
In the following we will focus on transmon multiqubit systems, and then the Hamiltonian takes the general form (omitting the harmonic oscillator term):
(13) 
For simplicity, in Eq. 4.3 the qubitresonator term is considered only to refer to readout and bus operations, leaving indirect qubitqubit interaction via the resonator to be included in via the coupling constant .
4.3.1 Capacitive coupling
4.3.2 Resonator coupling
In this case (Fig. 6c) the coupling is primarily indirect, via virtual excitation (polarisation) of the detuned bus resonator. Diagonalisation of the Hamiltonian gives the usual secondorder qubitqubit coupling [21, 24, 55]:
(17) 
(18) 
Here and are the detunings of the qubits, and . Finally, in the RWA one again obtains the JaynesCummings Hamiltonian
(19) 
4.3.3 Josephson junction coupling
The transmon qubits can also be coupled via a Josephson junction circuit [55], as illustrated in Fig. 6d. A case of direct JJcoupling (omitting the coupling capacitors in Fig. 6d) has recently been treated theoretically and implemented experimentally by Martinis and coworkers [151, 218]. To a good approximation, the Hamiltonian is given by
(20) 
(21) 
where the approximate result for refers to identical qubits in resonance, and the RWA Hamiltonian is again given by Eq. 16. By varying the flux in the coupling loop, the Josephson inductance can be varied between zero and strong coupling, [151].
4.3.4 Tunable coupling
Tunable qubitqubit coupling can be achieved in a number of ways, for example (i) by tuning two qubits directly into resonance with each other; (ii) by tuning the qubits (sequentially) into resonance with the resonator; (iii) by tuning the resonator sequentially and adiabatically into resonance with the qubits [219]; (iv) by driving the qubits with microwave radiation and coupling via sidebands; (v) by driving the qubit coupler with microwave radiation (e.g. [220]); (vi) by fluxtunable inductive (transformer) coupling [221]. In particular, for JJcoupling, the qubitqubit coupling can be made tunable by currentbiasing the coupling JJ [218, 222, 223].
5 Hybrid circuits and systems
In this section we will discuss the status of the DiVincenzo criteria DV6 and DV7 listed in Sect. 3.1.
Even if a QIP system in principle can consist of a single large coherent register of qubits, practical systems will most likely be built as hybrid systems with different types of specialized quantum components: qubits, resonators, buses, memory, interfaces. The relatively short coherence time of JJqubits () compared to spin qubits () and trapped ions () has promoted visions of architectures with fast shortlived JJqubit processors coupled to longlived memories and microwaveoptical interfaces, as illustrated in Fig. 7.
There are numerous demonstrations of coherent transfer between JJqubits and microwave resonators (both lumped circuits and microwave cavities, and mechanical resonators), as well as between JJqubits and spin ensembles. In principle, qubits coupled to microwave resonators (qcQED) is already a hybrid technology. An interesting aspect is that the development of longlived transmon qubits and highQ 2D and 3D resonators has changed the playground, and it is no longer clear what other kind of hybrid memory devices are needed for shortterm quantum memory. Even for longterm memory, the issue is not clear: with emerging quantum error correction (QEC) techniques it may be possible to dynamically ”refresh” JJcQED systems and prolong coherence times at will. To achieve longterm ”static” quantum memory, spin ensembles are still likely candidates, but much development remains.
The current situation for hybrid systems is described in three excellent review and research articles [224, 225, 226]. Here we will only briefly mention a few general aspects in order to connect to the DiVincenzo criteria DV6 and DV7, and to refer to some of the most recent work.
5.1 Quantum interfaces for qubit interconversion (DV6)
The name of the game is to achieve strong coupling between elementary excitations (e.g. photons, phonons, spin waves, electrons) of two or more different components so that the mixing leads to pronounced sideband structures. This can then be used for entangling different types of excitations for information storage or conversion from localised to flying qubits.
5.1.1 TransmonspincQED
Experimentally, strong coupling between an ensemble of electronic spins and a superconducting resonator (Fig. 7) has been demonstrated spectroscopically, using NV centres in diamond crystal [227, 228, 229] and spins doped in a [230].
Moreover, storage of a microwave field into multimode collective excitations of a spin ensemble has recently been achieved [231, 232]. This involved the active reset of the nitrogenvacancy spins into their ground state by optical pumping and their refocusing by Hahnecho sequences. This made it possible to store multiple microwave pulses at the picowatt level and to retrieve them after up to 35 , a three orders of magnitude improvement compared to previous experiments [232].
The ultimate purpose is to connect qubits to the superconducting resonator bus, and to use the spin ensemble as a longlived memory. Such experiments have been performed, entangling a transmon with a NV spin ensemble [233] via a frequencytunable superconducting resonator acting as a quantum bus, storing and retrieving the state of the qubit. Although these results constitute a proof of the concept of spinensemblebased quantum memory for superconducting qubits, the lifetime, coherence and fidelity of spin ensembles are still far from what is needed. Similar results were also achieved by directly coupling a flux qubit to an ensemble of NV centers without a resonator bus [234].
Finally, strong coupling between a transmon qubit and magnon modes in a ferromagnetic sphere has recently been achieved [235, 236], demonstrating magnonvacuuminduced Rabi splitting, as well as tunable magnonqubit coupling utilising a parametric drive. The approach provides efficient means for quantum control and measurement of the magnon excitations and opens a new discipline of quantum magnonics.
5.1.2 Transmonmicromechanical oscillatorcQED
Mechanical oscillators (Fig. 7) can be designed to have resonance frequencies in the microwave range and achieve strong coupling to superconducting qubits. Mechanical resonators therefore provide a new type of quantum mode  localised phonons. However, for this to be useful for quantum information processing one must be able to cool the mechanical oscillator to its ground state, to be able to create and control single phonons [237, 238]. It is then possible to induce Rabi oscillations between the transmon and the oscillator by microwave driving via motional sidebands, resulting in periodic entanglement of the qubit and the micromechanical oscillator [239].
5.1.3 TransmonSAW
Surface acoustic waves (SAW) are propagating modes of surface vibrations  sound waves. Recently, propagating SAW phonons on the surface of a piezoelectric crystal have been coupled to a transmon in the quantum regime (Fig. 8), reproducing findings from quantum optics with sound taking over the role of light [240]. The results highlight the similarities between phonons and photons but also point to new opportunities arising from the unique features of quantum mechanical sound. The low propagation speed of phonons should enable new dynamic schemes for processing quantum information, and the short wavelength allows regimes of atomic physics to be explored that cannot be reached in photonic systems [241].
5.1.4 TransmonHBAR
The recent development of bulk acoustic resonators [237, 244] has made possible the experimental demonstration [245] of a high frequency, highQ, bulk acoustic wave resonator (highovertone bulk acoustic resonator) (HBAR) that is strongly coupled to a superconducting transmon qubit using piezoelectric transduction. The system was used to demonstrate basic quantum swap operations on the coupled qubitphonon system. The relaxation time of the qubit was found to be 6 . Moreover, and for the lowest phonon level in the resonator was found to be 17 and 27 resp. The analogy to 3D cQED is obvious, but the thickness of the device is only about 0.5 mm, so it looks more 2D than 3D. It is expected [245] that fairly straightforward improvements will make cavity quantum acoustodynamics (cQAD) a novel resource for building scalable hybrid quantum systems.
5.2 Quantum interfaces to flying qubits (DV7)
The principle is that of good old radio technology: from the transmitter side, one achieves lowfrequency () modulation of a strong highfrequency () carrier (pump) beam by controlling the amplitude, frequency or phase of the carrier. The modulation is achieved by mixing the signals in a nonlinear device, creating sidebands around the carrier frequency.
In the present case, the mixers are different types of electrooptomechanical oscillators that influence the conditions for transmitting or reflecting the optical carrier beam. Typically three different oscillators are coupled in series, as illustrated in Fig. 7: a microwave resonator (), a micro/nanomechanical oscillator (), and an optical cavity (), with coupling energies and , resulting in the following standard Hamiltonian:
(22)  
The mechanical oscillator changes the frequency of the optical cavity. This is the same principle as readout: the phase of the reflected carrier carries information about the state of the reflecting device. Here the phase of the reflected optical beam maps the state of the mechanical oscillator. Tuning the laser frequency so that , either sideband is now in resonance with the optical cavity. If the resonance linewidth of the optical cavity is smaller than , then the sideband is resolved and will show a strong resonance. Adding a (transmon) qubit coupled to the microwave resonator (Fig. 7) one then has a chain of coupled devices that, if coherent, can entangle the localised qubit with the optical beam and the flying photon qubits.
To create this entanglement is clearly a major challenge, and coherent coupling has so far only been achieved to varying degrees between various components. We will now briefly describe a few technical approaches to the central oscillator components: piezoelectric optomechanical oscillator [246], micromechanical membrane oscillator [247, 248, 249, 250], collective spin (magnon) oscillator[251, 252, 253, 235, 236], and SAW [243].
5.2.1 Microwaveoptical conversion: optomechanics
This approach (Fig. 9) is based on the established optomechanical devices for modulating light [224], and has been investigated experimentally [246]. A beam of piezoelectrical material is patterned to contain a nanophotonic (1D) crystal, localizing light in a region of enhanced vibrational amplitude.
5.2.2 Microwaveoptical conversion: micromechanics
This approach (Fig. 10) is based on the wellknown technique of modulating reflected light, e.g. to determine the position of the tip of an AFM probe. The radiation pressure (light intensity) exerts a ponderomotive force on the membrane (Fig. 10), coupling the mechanical oscillator and the optical cavity [248]. There are proofofconcept experimental results showing coherent bidirectional efficient conversion of microwave photons and optical photons [247]. Moreover, this technique was recently used for demonstrating optical detection of radiowaves [249].
5.2.3 Microwaveoptical conversion: cavity optomagnonics
The Nakamura group has been investigating the coupling of microwave photons to collective spin excitations  magnons  in a macroscopic sphere of ferromagnetic insulator [251, 252, 253, 235, 236]. They recently demonstrated strong coupling between single magnons in a magnetostatic mode in the sphere and a microwave cavity mode [251, 252], including bidirectional conversion [253].
5.2.4 Microwaveoptical conversion: SAW
Shumeiko [243] has presented a theory for a reversible quantum transducer (Fig. 11) connecting superconducting qubits and optical photons using acoustic waves in piezoelectrics. The approach employs stimulated Brillouin scattering for phononphoton conversion, and the piezoelectric effect for coupling of phonons to qubits. It is shown that full and faithful quantum conversion is feasible with stateoftheart integrated acoustooptics.
6 Quantum gates
6.1 Quantum state time evolution
In quantum information processing (QIP) one maps classical data on the Hilbert space of a given quantum circuit, studies the resulting time evolution of the quantum system, performs readout measurements of quantum registers, and analyses the classical output. At this level there is no difference between quantum computing (QC) and quantum simulation (QS).
Timeevolution operator
The time evolution of a manybody system can be described by the Schrödinger equation for the state vector ,
(23) 
in terms of the timeevolution operator
(24) 
determined by the timedependent manybody Hamiltonian of the system
(25) 
describing the intrinsic system and the applied control operations. Gates are the results of applying specific control pulses to selected parts of a physical circuit. This affects the various terms in the Hamiltonian by making them timedependent.
For simplicity, can be regarded as timeindependent, and taken to describe DC and microwave drives controlling the parameters of the total Hamiltonian. This involves e.g. tuning of qubits and resonators for coupling and readout, or setting up and evolving the Hamiltonian. In addition, can introduce new driving terms with different symmetries. In general, the perturbing noise from the environment can be regarded as additional time dependence of the control parameters.
For the transmon, the system Hamiltonian takes the form in the RWA (same notation as in Sect. 4.3):
(26)  
(27) 
and the control term can be written as
(28) 
The time dependent allows switching on and off or modulating the various terms in , as well as introducing pulse shapes for optimal control. In Eq. (28), the first term provides general types of singlequbit gates, the second term describes qubitqubit coupling explicitly introduced by external driving, and the third term tuning of the frequency of the oscillator. Moreover, in Eq. (28), the first term allows tuning of the qubit energies in and out of resonance with the oscillator, making it possible to switch on and off the qubitoscillator coupling as well as creating oscillatormediated qubitqubit coupling. In the same way, the third term makes it possible to tune the oscillator itself in and out of resonance with the qubits.
The solution of Schrödinger equation for may be written as
(29) 
and in terms of the timeordering operator :
(30) 
describing the time evolution of the entire manyparticle state in the interval . in Eq. 30 is the basis for describing all kinds of quantum information processing, from the gate model for quantum computing to adiabatic quantum simulation. If the total Hamiltonian commutes with itself at different times, the time ordering can be omitted,
(31) 
This describes the timeevolution controlled by a homogeneous timedependent potential or electromagnetic field, e.g. dc or ac pulses with finite rise times, or more or less complicated pulse shapes, but having no spacedependence. Moreover, if the Hamiltonian is constant in the interval , then the evolution operator takes the simple form
(32) 
describing stepwise timeevolution.
Computation is achieved by sequentially turning on and off 1q and 2q gates, in parallel on different groups of qubits, inducing effective qubit gates.
6.2 Gate operations
The timedevelopment will depend on how many terms are switched on in the Hamiltonian during a given time interval. In the ideal case all terms are switched off except for those selected for the specific computational step. A single qubit gate operation then involves turning on a particular term in the Hamiltonian for a specific qubit, while a twoqubit gate involves turning on an interaction term between two specific qubits. In principle one can perform direct qubit gate operations by turning on interactions among all qubits.
6.3 1q rotation gates
1q gates are associated with the timedependent 1q term of the control Hamiltonian: . Expanding the state vector in a computational 1q basis, one obtains for a given single qubit,
(33) 
For a general control Hamiltonian the operators do not commute, and the exponential cannot be factorised in terms of products of , and terms. To get a product we must apply the operators sequentially, acting in different time slots. In that case, for a given operator we get
(34) 
where = = .
Expanding the exponential, calculating the matrix elements, and resumming, one obtains the time evolution in terms of rotation operators :
(35) 
where
(36) 
(37) 
(38) 
describing single qubit rotations around the x, y, and zaxes.
6.4 2q resonance gates
6.4.1 iSWAP
The 2q iSWAP gate can be implemented by using for tuning the energy of one of the qubits onto resonance with the other qubit, thereby effectively turning on the qubitqubit interaction in .
Expanding the state vector in a computational 2q basis, one obtains
(39) 
If the qubits are on resonance (), then the matrix elements of the 2qubit interaction part of the time evolution operator take the form
(40)  
Expanding the exponential function, introducing the matrix elements and resumming yields
(41) 
(42) 
referred to as the iSWAP gate.
Creating an excited state from by a pulse, the iSWAP gate describes how the system oscillates between the and states. The gate is obtained by choosing ,
(43) 
putting the system in a Bellstate type of superposition .
6.4.2 Cphase
The CPHASE gate can be implemented by making use of the spectral repulsion from the third level of the transmon, as shown in Fig. 12. One first uses twocolour pulses to drive both qubits from to , inducing a transition (point I). Then tuning one of the qubits rapidly into (near) resonance with the other one, staying at the crossing of the and levels for a certain time (point II), produces an interactiondependent shift of the level relative to . This induces an iSWAP gate between and .
To see this, one expands the state vector in an extended computational basis:
(44) 
and calculates the matrix elements (like in Sect. 6.4.1). The resulting energy level spectra in Fig. 12 show an avoided level crossing and a frequency shift of the level due to repulsion from the level, as shown in detail in Fig. 12b.
In this representation the evolution operator is diagonal, with the result that
(45) 
with
(46) 
(47) 
In the experiment, the 1102 splitting is determined by the timedependent bias tuning voltage in Fig. 12. If (point II), then
(48) 
After time t such that
(49) 
then the 11 state has rotated twice, and the phase is given by .
(50) 
At this point, the excursion of the bias voltage will decide the integrated strength needed for achieving , providing the CPHASE gate (Fig. 13a):
(51) 
6.4.3 Cnot
The CNOT gate can be expressed in terms of CPHASE and two Hadamard gates, as commonly implemented in transmon circuits (Fig. 13b):
(52) 
The first Hgate changes from the z to the xbasis, and the second Hgate transforms back.