Reviews in Physics 3 (2018) 1–14

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Reviews in Physics

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Analogue simulation with the use of artiﬁcial quantum coherent structures T ⁎ A.M. Zagoskin ,a,b a Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom b Department of Theoretical Physics and Quantum Technologies, National University of Science and Technology MISIS, 4 Leninsky Ave., Moscow 119049, Russia

ABSTRACT

An explosive development of quantum technologies since 1999 allowed the creation of arrays of natural and artiﬁcial quantum unit elements (viz. trapped ions and superconducting qubits), which maintain certain degree of quantum coherence and allow a degree of control over their quantum state. A natural application of such structures is towards simulating quantum systems, which are too big or too complex to allow a simulation with the means of classical computers. A digital quantum simulation promises a controlled accuracy, scalability and versatility, but it imposes practically as strict requirements on the hardware as a universal quantum computation. The other approach, analogue quantum simulation, is less demanding and thus more promising in short-to-medium term. It has already provided interesting results within the current experimental means and can be used as a stopgap approach as well as the means towards the perfecting of quantum technologies. Here I review the status of the ﬁeld and discuss its prospects and the role it will play in the development of digital quantum simulation, universal quantum computing and, more broadly, quantum engineering.

1. Introduction

In their paper [1] Dowling and Milburn proclaimed the advent of the Second Quantum Revolution and pinned its true beginning on 1994, when the Shor algorithm for breaking the RSA cryptosystem was discovered [2] and quantum entanglement between photons was demonstrated over 4 km of optical-fiber [3]. The first of these breakthroughs threatened one of the cornerstones of secure data transmission, on which much of the world economy came to depend in a short time; the second provided means of avoiding this danger; this naturally made the topic attractive to both researchers and funding bodies and spurred the further development of the field. One could take instead the year 1999, when the first superconducting qubits were realized (charge [4] and flux [5]) and D-Wave Systems Inc. was founded [6], but this would not change much. As pointed out in [1], “in the Second Quantum Revolution we are now actively employing quantum mechanics to alter the quantum face of our physical world”. But, to a degree, this can be also said about the First Quantum Revolution (which can be symbolically pinned on 2 December 1942, when the first nuclear reactor achieved criticality), which encompassed nuclear power, semiconductor devices, lasers and superconductors. The key difference between the two is rather that the First Revolution and the corresponding “Quantum Technologies 1.0” employed quantum effects on microscopic scale only, that is, the quantum states of the corresponding systems included quantum superpositions and entanglement of only a small number of microscopic quantum states

⁎ Corresponding author at: Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom. E-mail address: [email protected]. https://doi.org/10.1016/j.revip.2017.11.001

Available online 15 November 2017 2405-4283/ © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/). A.M. Zagoskin Reviews in Physics 3 (2018) 1–14 across microscopic distances. For example, the order parameter, which characterizes a superconductor, is often described as its “macroscopic wave function”, but it has just a single degree of freedom, like a single quantum particle. Quantum superpositions involved in the operation of a dc or rf SQUID in its standard regime, as described in, e.g., [7], only concern quantum states differing by a single Cooper pair per Josephson junction (see [8], Section 3). “Quantum Technologies 2.0” (QT2.0), on the other hand, use quantum superpositions and/or entanglement of quantum states of artificial structures on a macroscopic (or at least mesoscopic) scale. This definition should not be applied too strictly. For example, quantum communications do not require entanglement between a macroscopic number of photons, though its spatial scale must be macroscopic. Superconducting or semiconductor charge qubits for their operation require quantum superpositions of states differing only by a single Cooper pair or electron (while in superconducting flux qubits the difference is at least 10105 − 6 single particle states) [9]. Qubits based on nitrogen-vacancy (NV−) centres in diamond [10,11], divacancies in silicon carbide [12] or two-level systems in tunneling barriers of Josephson junctions[13,14] exploit microscopic degrees of freedom of natural systems. Still, such a definition is convenient, clear enough and gives the gist of what the Second Quantum Revolution and QT2.0 are about. The main directions of development of QT2.0 outlined in 2002 [1] did not essentially change by now. The 2014 report by the UK Defense Science and Technology Laboratory [15], the 2016 UK Government Blackett report [16] and the 2016 EU Quantum Man- ifesto [17] list such applications as quantum clocks, quantum imaging, quantum sensing and measurement, quantum computing and simulation, and quantum communication, and estimate that it will take from 5 to 15 plus years for different sections of QT2.0 to be realized. Similar time scales are indicated in the Innovate UK report on commercialization of quantum technologies in the finance sector [18]. It is also predicted [17] that by 2035 a quantum computer will exceed the power of conventional computers. The global market for different quantum technologies is estimated at anything from 1–10 million to 0.1–10 billion pounds (for the long-term), and is, of course, predicated on the absence of any fundamental barrier on the way towards the realization of large-scale quantum coherent systems (e.g., due to a gravitational quantum state reduction [19]). So far no indication of such a barrier was detected. The prevailing classification of QT2.0 by application is not the most logical approach from the point of view of physics, but it reflects the important fact that the development and investigation of such quantum structures takes on features of engineering, when for the first time quantum coherence, entanglement and superposition must be treated as material parameters determining the performance, on par with Young’s modulus or electrical resistivity. The engineering aspects of QT2.0 pose formidable challenges. For example, it is necessary to maintain quantum coherence of the system (which demands its insulation from the environment) while simultaneously manipulating its quantum state (which a con- nection to the environment and extra circuitry being an additional source of decoherence). This is a typical engineering task of meeting mutually incompatible requirements. There is also a novel challenge, the fundamental impossibility of modeling large enough quantum structures with classical means, pointed out by Feynman as the motivation for developing quantum computing (see, e.g., [20]). At the moment the limit is (very optimistically) about a hundred qubits, and it is unlikely that it can be pushed much farther. On the other side, a universal quantum computer should contain thousands of logic qubits (and at least an order of magnitude more physical qubits), far exceeding this limit. This makes the development of new methods specific for quantum engineering an urgent and challenging task [21]. Analogue quantum simulation and computation is one way of meeting these challenges. Analogue devices operate by a direct simulation: they are designed in such a way that their natural evolution is described by approximately the same equations as that of the simulated system. For example, slide rules used the rigidity of solids to calculate ln(ab )=+ ln a ln b and specially designed scales to realize the input and output, a → ln a, b → ln b,ln(ab) → ab. They are by definition not universal, unlike digital computers, and must be specifically designed for a certain class of problems. Their accuracy is limited by the accuracy of input and readout and by the intrinsic accuracy of its operation, i.e., by how well the natural evolution of the analogue device approximates that of the simulated system. In the case of a slide rule, the former is determined by the width of the marks on the scales and the cursor, and the latter by the accuracy of scales and the rigidity of the structure. The accuracy of an analogue quantum computer is thus hardware-determined and cannot be increased at will. On the other hand, its operation being the natural evolution, it does not demand a precise time- domain control of its individual elements, and being designed for a specific task, it can benefit from problem-specific design shortcuts. This loosens the requirements to the analogue quantum hardware and allows to simplify the design, thus reducing the scale and complexity of the device and with it the sources of decoherence, compared to a digital quantum computer. The use of analogue quantum devices for quantum optimization, modeling complex natural processes and the evaluation of artificial quantum structures brings additional benefits of perfecting the quantum technologies themselves. In the following we will review these applications, realizations of quantum analogue devices, and possible directions of their further development. This is a very quickly developing area of research, and here we can only cover the indicative directions, in order to give the reader a general overview. We will concentrate on the most advanced quantum hardware, which combines the control over individual qubits with the best promise of scalability. Arguably, these include superconducting quantum structures [22,23], trapped ions [23,24], optical lattices [25,26] and spins in solid state (e.g., NV−-centres in diamond [10,11]).

2. Digital quantum simulation

Simulation of quantum systems with the use of artiﬁcial quantum structures is arguably the most natural application of QT2.0 and is a popular and quickly expanding ﬁeld of research [27,28]. In the most direct approach, given the initial state of a quantum system, |ψ(0)⟩, and its Hamiltonian Ht (), one would reproduce the unitary evolution of the system via simulating the action of the evolution

2 A.M. Zagoskin Reviews in Physics 3 (2018) 1–14 operator,

 ⎛ t  ⎞ ψt()=≡ Ut () ψ (0)T exp⎜⎟−i∫ H ( t′′ ) dt ψ (0). ⎝ 0 ⎠ (1) In case of non-unitary evolution, one needs to simulate the Liouville-von Neumann equation for the density operator ρ ̂(),t

∂ttρt̂()= L [ ρ ], ̂ (2) where the Liouville superoperator in most relevant cases can be chosen in the Markovian approximation, ̂  ̂ ̂̂ †† ̂ ̂̂ ̂ Lt [ρ ]=− iHt [ ( ), ρt ( )]+∑ (2 LρtLa ( )aa− { LLa , ρt ( )}). a (3) ̂ Here La are the Lindblad operators, which determine the non-unitary evolution. Lloyd [29] developed the general scheme of universal quantum simulation based on the Trotter formula (Ref. [30], 4.7.2), lim()eeiA t// n iB t n n= e i () A+ B t , n→∞ (4) and its modiﬁcations, which yield

eeeOteeeOti( A+ B )Δ t=+= iA Δ t iB Δ t (Δ2 )iA Δ/2 t iB Δ t iA Δ/2 t + (Δ 3) (5) etc. Operators AB,  generally do not commute and may be non-Hermitian, which allows using the approach for nonunitary evolution. The scheme [29] applies to systems with Hamiltonians, which can be represented as a sum of terms, each acting on only a ’  subspace of the system s Hilbert space, Ht()= ∑k Hk ( t). Most practically interesting systems fall in this category. Slicing the time interval t into intervals ΔttN= / and applying the ﬁrst-order Trotter approximation to the evolution operator Ut ()(1),weﬁnd that up to O (Δ)Nt22= OtN (/)

 Ut ()≈≈∏∏ e−−iH()Δ tjk t ⎛ ∏ eiH ()Δ tj t⎞. j jk⎝ ⎠ (6) The number of steps necessary to achieve accuracy ϵ is N ∼ t2/ϵ (or, restoring the dimensionful constants, N ∼ (Et/ℏ)2/ϵ, where E is the characteristic energy scale). Higher-order Trotter approximations can provide better results [27,30], but the necessity of a large number of elementary unitary operations performed on the system remain. The great advantage of this strategy of digital quantum simulation is its universality. A straightforward extension of the above scheme using ancillary qubits, which play the role of the environment, allows to simulate nonunitary evolution [29]. The algorithm and protocol for such simulation were developed in Ref. [31]. Other schemes were proposed as well [32–34]. Moreover, an arbitrary quantum many-body system can be simulated using an array of qubits with appropriate two-qubit interactions; in particular, XY- and Heisenberg spin-lattice models are “universal quantum simulators” [35]. Since an arbitrary evolution of a n-qubit system can be reduced to a sequence of one-qubit rotations and two-qubit gates, the evolution operator (6) with an arbitrary many-body Ha- miltonian can be simulated using an array of qubits with a universal set of gates. The results which can be obtained in this way are not limited to a straightforward simulation of quantum evolution. For example, there was proposed an efficient method of obtaining n-time quantum correlators [36] applicable to spin, Fermi and Bose operators. Another example is an image processing algorithm, where an image is mapped on a statistical operator, the latter is evolved according to (2) with an appropriate choice of Lindblad operators, and then converted back to the image [37] (Fig. 1). Even a numerical simulation of this process demonstrates a significant and efficient noise reduction. A special interest is presented by quantum chemical problems, which pose a difficult task for classical computers, and where a small number of qubits is sometimes sufficient. Then one can use NMR quantum computing technique [38], capitalizing on the fact that in this approach one can perform a long sequence of precise qubit manipulations. In quantum chemistry even finding the ground or low-energy state configuration of a system is a challenge; therefore quantum adiabatic optimization with a small number of qubits may prove useful as well. Liquid-state NMR can manipulate a dozen qubits [39] and is not likely to greatly exceed this number. Still,

Fig. 1. Noise reduction with a quantum image processing algorithm [37]. From left to right: original image, image with added noise, image after the application of a quantum image processing algorithm with diﬀerent processing times (numerical simulation).

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Fig. 2. Digital quantum simulation using liquid NMR technique with 4 qubits (13C spins in a molecule of trans-crotonic acid) [41]. A frustrated three-spin system in the external magnetic field is simulated. The total magnetization (open red circles) is in a good agreement with numerical calculations (blue crosses). The number of applied one- and two-qubit gates is over 100. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) computations were performed with this technique (e.g., Fano-Anderson model [40] (using 7 spin-1/2 qubits in a molecule of trans- crotonic acid), frustrated state of three Ising spins [41] (4 qubits in the same molecule, see Fig. 2), ground state energy of hydrogen molecule [42] (2 qubits in a CHCl3 molecule)). In order to increase the number of qubits in NMR approach without an exponential suppression of signal-to-noise ratio one has to switch to spins in solid state (e.g., NV−-centres [10,11]), where other methods of qubit control may be preferable. In another experiment digital quantum simulation was realized in a chain of trapped ions [43]. The simulator contained 6 qubits + (represented by D5/2 and S1/2 Zeeman states of Ca ions) interacting via a vibration mode of the ion string. In the experiment the system of two spins with Ising (Fig. 3), XY and XYZ interactions, three-, four- and six-spin Ising systems and a six-spin system with six- body interaction were simulated. This required up to 100 gates. Trapped ions in general provide a convenient tool for digital quantum simulation due to the long decoherence times and good control overe quantum states. For example, controlled dissipative quantum dynamics was realized using 4+1 40Ca+ ions (the one extra qubit playing the role of the environment) [44]. It was used for the dissipative preparation of a 4-qubit GHZ state ( 0000+ 1111 )/ 2 with the probability of ∼ 0.8 after three operation cycles. Each cycle includes 16 five-ion entangling operations, 20 collective unitary operations and 34 single-qubit operations [44]; this gives an idea of how involved are the manipulations with qubits required for digital quantum simulations in general.

Fig. 3. Digital quantum simulation with trapped ions [43]. Evolution of two simulated Ising spins in a time-independent external magnetic ﬁeld is shown. The number of simulation steps 5, 10, 15, 20 (top to bottom); solid lines indicate the exact solution.

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Wecker et al. [45] estimated that in order to compete with the current classical machines a quantum computer running digital quantum simulation needs to use at least 50 spin orbitals (i.e., at least 50 logic qubits), and at least 100 in order to have a significant advantage. Reaching this scale is not impossible, but the need of performing a long sequence of precise qubit manipulations pre- scribed by the Trotter formula (4) and its generalizations would require the use of quantum error correction, thus increasing the number of physical qubits by at least an order of magnitude. This is considered the main problem of digital quantum simulation: as estimated in Ref. [45], optimistically the number of steps scales as O(N9) (as O(N8) if use parallelization). Here an additional difficulty is presented by Fermi statistics, where one has to rely on Jordan-Wigner transformation [46] to represent fermionic degrees of freedom. Fortunately, it was shown that it is possible to reduce the corresponding gate count overhead from O(N)toO(1), parallelize gate application and increase the Trotter step [47]. Nevertheless the number of steps will likely to scale no better than O (N6), which puts a ‘competitive’ quantum simulator beyond the reach of current quantum hardware. The downside of digital quantum simulation is the extension of its advantages: such a simulator is in fact a universal quantum computer, with the correspondingly high demands to its hardware. It is therefore reasonable to consider non-universal approaches, which can be realized with currently available artificial quantum devices. One such approach is analogue quantum simulation, where we take advantage of the coincidence or proximity of equations governing the dynamics of the simulator and the system to be simulated [27,28]. The obvious advantage of this approach is that one can use the natural evolution of the simulator, thus reducing the operations on qubits to tuning the parameters and initializing the device before each run. This allows to greatly simplify the design of the simulator and minimize its interactions with the environment during the run, thus suppressing both intrinsic and external sources of decoherence. The disadvantage is the extension of the advantages: there is no universality, and the precision of the computation is limited. Nevertheless a slide rule in hand is better than a laptop in the bush.

3. Analogue simulation with small quantum structures

One of the smallest structures usable as a quantum simulator a single ion. In Ref. [48] a trapped 40Ca+ ion was used to simulate the Dirac equation in 1+1 dimensions:

2 i∂=txzψcσmcσψ()p̂ + . (7)

The states Sm1/2,1/2J= and Dm5/2,3/2J= were uses as spinor states, transitions between them were controlled by a laser field. The Hamiltonian of the trapped ion in the presence of light fields, ∼ HησσDxz=+2ΔΩp̂ Ω , (8) mimics the Dirac Hamiltonian of (7). Here Δ is the size of the ion wave packet in the trap, p̂=−ia()/2Δ† a is the momentum operator ∼ along the trap axis, η the Lamb-Dicke parameter, and Ω and Ω determined by the intensity and frequencies of the light field. This ∼ means that in the simulation the “speed of light”, cη͠ = 2ΩΔ, and the “particle mass” can be varied. The simulation allowed to reproduce the Zitterbewegung as a function of the particle mass (Fig. 4). A system of two trapped ions was used to simulate two quantum spins [49], and a chain of three trapped 171Yb+ ions allowed to simulate a frustrated three-spin quantum network [50] (Fig. 5). In the Ising Hamiltonian

Fig. 4. Analogue quantum simulation of the Dirac equation in 1+1 dimensions with a single trapped ion [48]. The expectation value of the particle position vs. time is −1 plotted for different effective masses. The “speed of light” is cμ͠ = 0.052Δ s . The “Compton wavelength” λC = c͠ /Ω is (top to bottom curve) 0.6Δ, 1.2Δ, 2.5Δ, 5.4Δ, and infinity (the uppermost curve corresponds to a massless particle). As the mass of the particle decreases, the Zitterbewegung becomes less pronounced. Solid lines represent numerical simulations.

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Fig. 5. Analogue quantum simulation of an Ising magnet [50] with three trapped 171Yb+ ions. Populations of diﬀerent states (symbols) are measured after the Ising ∼ interactions are suddenly switched on and the transverse magnetic ﬁeld is being adiabatically lifted. The Ising couplings, Jij, are determined by the laser detuning μ . Here all interactions are antiferromagnetic (b) or ferromagnetic (c). The theoretical curves take into account non-adiabaticity (solid lines) compared to the exact ground state (dotted lines).

 i j i HJσσBσ=+∑∑ij x x y y ij< i (9) the coupling parameters, determined by the Coulomb interaction between the ions, could be changed by detuning the lasers from spin resonance. The measurements were performed by first initializing the electronic states of ions, then applying the Ising coupling and a strong transverse magnetic field, and finally adiabatically lifting the field. It is interesting to compare this approach with the digital approach of [41]. While the accuracy of the latter is better, the procedure in Ref. [50] is simpler and allows a straightforward extension to a larger number of qubits. Analogue ion trap techniques allow to simulate nonequilibrium dynamics of quasiparticle and entanglement propagation in a system described by the Ising Hamiltonian (9). Jurcevic et al. (Ref. [51]) used for this purpose 1D chains of seven and fifteen 40Ca+ ions. Flipping one or several spins allowed to send quasiparticle excitations along the chain and observe the propagation of quasi- particles and of quantum correlations. The coupling strength had an approximately power-law dependence on intersite distance, −α Jij ∝−ij, with the exponent tunable between α = 3 (short range) and α = 0 (infinite-range interaction). The measurements, in particular, showed how the “light cone” structure of quantum dynamics following a local quench disappears as the interaction range increases (Fig. 6). Now consider a superconducting flux qubit placed inside a transmission line [52] (Fig. 7). The qubit interaction with the electromagnetic field in the line is analogous to that of a dipole atom-field interaction, which allows us to simulate a variety of quantum- optical effects. The qubit Hamiltonian

 Δ η iωt− iωt Hσ=−zx −σe( + e ) 22 (10) is analogous to that of the semiclassical theory of atom-field interaction, where the field is treated classically (Ref. [53], Ch.XV; Ref. 2 [9], 6.1.1). Here the coupling between the qubit and the field, η = MIp I0/ c , is determined by the mutual inductance M between the qubit and the line, and the corresponding current amplitudes, Ip and I0. The resonance frequency of the qubit is 10.204 GHz. The wavelength of the electromagnetic field at this frequency range, ∼ 3 cm, greatly exceeds the qubit size, ∼ 10μm. This allows us to treat the qubit as a point scatterer, and to write for the current in the line I(x, t)

−2 ̂ ∂xxIv−∂=tt Icδxϕt() ∂tt (). (11)

ikx ik x− iωt ̂ Here Ixt(, )=+ ( Ie0 Iesc ) e , v is the speed of light in the transmission line, and ϕMIσ= pxplays the role of the atomic dipole moment. The measured reflection amplitude r (defined via IrIsc =− 0) as a function of frequency and amplitude of the incident field was in a very good agreement with the theoretical predictions following from Eqs. (10) and (11), Fig. 8. In case of inelastic scattering, the characteristic Mollow triplet of resonance fluorescence is observed (Fig. 9). Beyond the limits of applicability of the two-level approximation (10), this system allows to simulate, e.g., an electromagnetically induced transparency [54], which involves three energy states of the flux qubit. The efficiency and clarity with which quantum-optical effects can be simulated by the qubit in transmission line are to a large degree due to the strong coupling between the qubit and the field (the effective mutual inductance M is enhanced by the large size of the qubit itself and the galvanic coupling between the line and the qubit, see Fig. 7B) and the restricted field volume in the 1D geometry. The effective “atom-field” coupling strength in such a system ranges from 0.005 to 1.34 (Ref. [55]), compared to 310× −7 for real atoms in the optical range in 3D (Ref. [9], Table 4.1).

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i Fig. 6. Quantum dynamics of spin in a 15-spin (trapped ion) chain following a local quench in the middle of the chain [51]. The magnetization σtz () is measured. The exponent α, which characterizes the spatial decay of spin-spin interactions, ranges from 1.41 (a, short range) to 0.75 (c, long range). The “light cone” structure of the perturbation propagation is clear in (a), but washed out in (b) and (c).

Fig. 7. (A) Artificial atom interacting with electromagnetic field. (B) Superconducting flux qubit in a transmission line. [52].

Superconducting quantum circuitry allows simulation of a number of fundamental effects. The key property behind it is the 2 feature of the Josephson junction, which can be considered as a tunable nonlinear inductance, LφJc()= c /(2cos), eI φ where Ic is the Josephson critical current and φ is the phase difference across the junction (see, e.g., Ref. [9], 2.1.2). The value of this inductance can be changed very fast [56]. Therefore in a transmission line closed by a SQUID the boundary condition, and the effective length of the line, can be changed very fast by changing the magnetic flux through the SQUID loop. This approach was used by Wilson et al. [57] to demonstrate the dynamical Casimir effect in the microwave range. The effect consists in the excitation of photons from the vacuum state by a mirror, which moves with a speed v comparable to the speed of light. In the experiment [57] the inductance of the junction changed sinusoidally with frequency fd within the range 8–12 GHz. The effective vc/ ͠ ratio reached approximately 0.25, where the speed of light in the line cc͠ ≈ 0.4 . As a result, the generation of photons from vacuum was detected in a good agreement with theory (Fig. 10). A logical extension of this approach leads to the recent proposals of simulating the Unruh effect [58] or the relativistic motion-generated entanglement between two qubits [59]. Another effect simulated in a small quantum circuit is the Dicke superradiance [60], the realization of the original Dicke’s two- spin Gedankenexperiment. Two transmon qubits are placed in a leaky 1D resonator (Fig. 11). They are initialized in a given quantum state and allowed to decay; the emitted power show a clear deviation of the collective spontaneous emission rate from that of independent qubits (Fig. 12), in a quantitative agreement with the theory.

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Fig. 8. Reﬂection amplitude for the elastic scattering of 1D microwave signal by a superconducting ﬂux qubit [52]. (A) Dependence on signal detuning from the qubit resonance frequency of 10.204 GHz. (B) Dependence on the signal strength: experiment (top) and theory (bottom).

Fig. 9. Resonance ﬂuorescence in a superconducting ﬂux qubit in a transmission line [52]. The spectral density of scattered signal is plotted as a function of detuning (A) at the incident microwave power of -112 dBm and (B) as a function of detuning and power. The solid line in (A) is the theoretical curve.

4. Analogue simulation with quantum arrays

In the end, simulation of many-body physics requires a transition to larger quantum structures. Here optical lattices may have an advantage due to the large number of atoms in them. For example, this approach allows to simulate the Zak phase (the Berry phase gained by the particle crossing the Brillouin zone), which characterizes the topological properties of a 1D solid [61]. The phase in question is given by the integral over the 1st Brillouin zone,

G/2 φZ =∂iuudk∫ kkk , −G/2 (12) and is revealed in, e.g., the properties of polyacetylene or linearly conjugated diatomic polymers. The 1D optical lattice of Ref. [61] contained a Bose-Einstein condensate of 87Rb atoms; it had two cites per unit cell and could mimic either structure depending on the choice of parameters. The Hamiltonian of the system reproduced the Rice-Mele Hamiltonian of a diatomic polymer [62],

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Fig. 10. Simulation of the dynamical Casimir effect [57]. Microwave photon flux density produced by changing the effective length of the transmission line (photons −1 −1 s Hz ) as the function of pump power, at the detuning f −=−fd 764 MHz.

Fig. 11. Realization of the Dicke Gedankenexperiment with two transmon qubits (yellow) connected to a leaky 1D cavity (blue) [60]. The average relaxation and pure dephasing rates of the qubits are 0.041 MHz and 0.25 MHz repectively, the qubit-resonator coupling strength 3.6 MHz, and the cavity decay rate 43 MHz. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

 †† †† HJabJabhcaabb=−∑∑(..)Δ()n n + ′ n n−1 ++n n −n n , n n (13)

† † Δ ﬀ where abnn, create atoms on the corresponding sublattice site. The staggering parameter is controlled by the phase di erence between the standing optical waves forming the lattice and determines the energy diﬀerence between the sites within the unit cell. If Δ = 0, the Zak phase equals π/2, but it drops to zero as Δ increases. The atoms were adiabatically evolved through the Brillouin zone by applying microwave pulses, and the Zak phase was extracted from the resulting Ramsey fringe, in a good agreement with theoretical predictions (Fig. 13) Interesting results were obtained in a simulation of a Hubbard model with interacting fermions [63], when system of 2310−× 5 40K atoms in a lattice is allowed to freely expand. Quantum dynamics of such a system cannot be calculated in higher dimensions than 1D. Therefore it came as a surprise that the observed expansion did not depend on whether the interaction was attractive or repulsive. The result was explained by an exact dynamical symmetry of the Hubbard model, revealed in the conditions of the experiment.

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Fig. 12. Spontaneous emission in the cavity [60] (see Fig. 11). Two transmon qubits connected to a leaky 1D cavity are initialized in the excited state. (a) Individual decay of qubits A, B (purple and green dots). (b) Collective decay of qubits A, B (blue dots). The lower panels show the emission power, the upper panels the deviation from the average single-qubit emission power (red dots). The blue line is a theoretical prediction. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 13. The Zak phase (topological phase in a 1D solid) dependence on the energy diﬀerence Δ between the sites in the unit cell. The experiment was conducted on a 1D optical lattice of 87Rb atoms condensate [61].

Optical lattice devices are proposed for simulating other many body effects, e.g., topological phase transitions in superradiance lattices [64,65]. Superconducting quantum structures provide a greater flexibility with coupling configurations, while they are reasonably scalable, currently extending to 2000 flux qubits in an adiabatic quantum processor [66] and 4300 qubits coherently coupled to a microwave resonator [67]. Together with a well-developed design, fabrication and measurement techniques this makes them a very attractive basis for analogue quantum simulations [28,68]. The downside is the still non-negligible dispersion of parameters. So far the most prominent application of large superconducting quantum arrays was in the field of quantum optimization [69].A quantum annealer, or quantum optimizer, is the simplest realization of a quantum computing device [70] which solves optimization problems. The cost function is identified with the ground state energy of the system, and the solution with the set of qubit states. An adiabatic quantum computer is usually described by the quantum Ising Hamiltonian (9). It is possible in principle to encode the solution to any problem solvable by a universal quantum computer in the ground state of a system of qubits [71], at a price of more complex qubit-qubit interaction. Nevertheless the Ising model is general enough, as it allows to formulate many NP-complete problems, including all of Karp’s 21 NP-complete problems [72]. (Any solution of an NP-complete problem can be verified in a

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Fig. 14. Schematics of D-Wave 2X quantum annealer (dots represent qubits, lines connnections) [22]. Each of 144 registers nominally contains 8 flux qubits (not all qubits and connections are operational). polynomial time (i.e., the number of required steps grows only as a power of the size of the input), but there is no known way of finding the solution in a polynomial time. There is no proof that such a way does not exist either. Karp’s 21 problems [73] is one of the first sets of naturally occurring problems the NP-completeness of which was proven.) The idea of the approach is based on (1) the possibility of the “ground state encoding” (i.e. choosing the coupling parameters in (9) in such a way that the ground state of the  Hamiltonian Hf (the Ising Hamiltonian with By=0) encoded the solution of the problem) and (2) the adiabatic theorem of quantum mechanics stating that a slowly perturbed quantum system will generally remain in its ground state. Therefore an isolated system with the Hamiltonian

  j Hλf() t=+−≡+− λtH () (1 λt ()) H0 λtH ()f (1 λt ()) B y∑ σy j (14)   will evolve from the easily reachable ground state of H0 to that of Hf , thus yielding the solution, if only λ˙ is small enough. The approach is immediately attractive, since it disposes of the fast and precise time-domain manipulation of qubit states, required in the standard gate-based quantum computing, replacing it with the fixed qubit-qubit couplings (something readily realizable with superconducting hardware [74] and a slow application (or rather, lifting) of a simple perturbation acting uniformly on all qubits). This swap of time for space greatly reduces the complexity of the structure and thus eliminates additional sources of decoherence. In principle, one can distinguish between adiabatic quantum computing proper, when the system is coherent and isolated and stays in its ground state; quantum annealing, i.e. quantum-assisted search for the ground state of an open, partially coherent quantum system; and approximate adiabatic quantum computing, where a low excited state of an open quantum system provides a good enough approximation to the exact solution. Quantum optimizers currently contain up to 2000 qubits (in the DWave2000 processor). The previous iteration of this device contained nominally 1152 qubits [22], Fig. 14. The typical evolution time of D-Wave processors (5-15 s) greatly exceeds the decoherence time of a single qubit (100 ns) used in these devices. Therefore they cannot operate as adiabatic quantum computers in the narrow sense. Questions to what extent these devices operate in quantum regime are still being actively discussed (see, e.g., Ref. [75] and references therein). Quantum coherent character of operation of a single 8-qubit register within such device is well established [76], but modelling fully quantum evolution of a hundred or more qubits with classical computers is beyond any realistic possibi- lities. Indirect methods (e.g., comparing the statistics of the performance of an adiabatic annealer don a large set of optimization problems with known solutions to that of different types of optimization algorithms [77,78]), do not allow an unequivocal interpretation [79] and are not scalable beyond several hundred qubits. At any rate, these methods are of a very limited use in application to engineering (design, characterization and optimization) of large quantum coherent structures. What is worse, currently there is no accepted general criterion of quantumness, which could be efficiently accessed on experiment and theoretically evaluated. It seems likely that such criteria are not universal and depend on the specifics of the structure and the tasks it is designed to perform. It is therefore necessary to find means of bridging the gap between

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Fig. 15. Levels of engineering. our current ability to simulate large quantum systems and the capacities of the smallest workable quantum computers, which hopefully will be able to eﬀectively do this job. This requires developing speciﬁc quantum engineering approaches a task, which can be illustrated on the example of quantum engineering of superconducting devices.

5. Problems and prospects

As pointed out in the Introduction, the complexity and scale of quantum coherent devices produced by the Second Quantum Revolution presents their designers with a difficult task of maintaining the fragile quantum correlations in a macroscopic environment, while making impossible their simulation using conventional computers. Therefore quantum engineering is by necessity developing in lockstep with the development of new methods of simulating large-scale, quantum coherent systems. The broadly stated goal of engineering is to build reliable structures out of unreliable unit elements (Fig. 15). Quantum unit engineering is mature. The quantitative theory of various types of superconducting qubits is well developed and is in an excellent agreement with the experiment, building on fifty years of science and technology of the Josephson effect. The evidence of this is the fast development of a number of improved qubit types and the above mentioned spectacular increase in their decoherence time. As long as a direct numerical simulation is possible, design, characterization and optimization of a superconducting quantum register is a routine task. Same applies to other types of quantum structures we have discussed. The difficulties start at the structural level. The goal of conventional structural engineering is to determine the properties of a complex structure based on the properties of its constituent elements, to evaluate and optimize the overall design ensuring that its performance will satisfy the set requirements. Quantum structural engineering pursues the same goal, with the fundamental complication that a direct simulation is impossible. This should not be apriori considered as an insurmountable obstacle: such simulations were impossible in conventional structural engineering as well until the arrival of high-performance computers, and it did not stop the development of complex structures, e.g., aircraft and first modern computers. In our case the solution could be based on the identification of efficient quantumness criteria, which would provide a probabilistic estimate of the systems performance, the development and application of generalized approximate methods of nonequilibrium, nonstationary many-body theory tailored to compute these criteria and taking into account essential quantum correlations. This can be only done in a close collaboration with experiment, the theory both providing guidance and obtaining important insights and feedback. We concede that it is impossible to simulate an individual instance of a large quantum device (e.g., a quantum annealer with a given set of interqubit couplings). But it is plausible that the essential properties of an ensemble of such systems will be reflected in certain higher-level, global characteristics (figures-of-merit), which are insensitive to details of a particular instance, computable by classical tools and accessible to experimental investigation. Drawing on the analogy with aero/hydrodynamics, such quantum Reynolds numbers could then be used to characterize and optimize a complex quantum coherent structure. What is lacking is the quantum scaling theory, which would identify such criteria and allow extending the data from small-scale experiments to larger-scale devices. There is no clear candidate for such a theory at the moment. Several approaches seem possible. One can try to generalize the methods of nonequilibrium, nonstationary theory of quantum many-body systems, such as the Keldysh formalism (see, e.g., Ref. [80], Ch.10). It is necessary to hold on the essential multipoint phase relations while keeping the number of variables low enough to be classically computable. What relations are essential will be dictated by the system. An example of a QT2.0 device system where such approach may succeed is provided by quantum metamaterials [81,82]. These artificial media only need to maintain quantum coherence during the time exceeding the traveral time of the electromagnetic signal, and do not require a precise control over all their unit elements. The limited character of the necessary quantum coherence and control required allows a simplified description. For example, such effects as controlled signal propagation [81], lasing [83,84] or superradiance [85] can be described by approximating the quantum state of the qubit system by a factorized two-component wave function,

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  T Ψ=⊗jψt (,)rrrj =⊗j ((,),(,)) utvtjj, (15) where the product is taken over all constituent qubits. In the continuum limit, when the period of the structure is much less than the signal wavelength, it reduces to a set of equations for ψ (,r t) reminiscent of Bogoliubov-de Gennes equations for a superconductor. Together with the equations for the electromagnetic field they yield the equations for the electromagnetic wave propagating in a medium with an effective refractive index dependent on the qubit quantum state (which in turn is affected by the electromagnetic field). This approach becomes inadequate when the entanglement between qubits is essential, but quantum metamaterials due to their relative simplicity are a good testing bed for developing more powerful theoretical tools. In conclusion: The essence of quantum technologies 2.0 is in the utilization of multiparticle quantum superpositions and entanglement. This makes impossible a straightforward quantitative modelling, characterization and optimization of devices, which can be fabricated and have already been fabricated using the existing experimental techniques. New theoretical tools must be developed to answer this need, and an engineering combination of theory, heuristics, rule-of-thumb estimates, and extrapolation from model experiments may provide the solution.

Acknowledgments

The author acknowledges the ﬁnancial support of the EPSRC (grant Testing quantumness: from artiﬁcial quantum arrays to lattice spin models and spin liquids, EP/M006581/1) and of the Ministry of Education and Science of the Russian Federation in the fra- mework of Increase Competitiveness Program of NUST MISiS (No. K2-2016-067).

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