28.1 Introduction

In this chapter, we discuss algorithmic and experimental aspects of quantum control of spin and pseudospin systems in view of realizing quantum algorithms or quantum simulations (1,2) at minimal cost, in particular in a minimum amount of time. For example, we will see that the time required for implementing a quantum module experimentally is a most natural measure of cost, whereas the number of standard elementary gates, that is, the network complexity, often does not allow for a simple one‐to‐one translation into the actual time complexity. Further typical cost functions may include relaxative losses or sensitivity to experimental imperfection. In view of future technologies, considerable recent progress of steering experimental quantum systems (3) is due to combining the tools of two mature research disciplines: (i) magnetic resonance (4) with its ample arsenal of methodology (5) for manipulating quantum systems and (ii) optimal control theory (6,7), nowadays an indispensable tool in system theory (8) and engineering (9). Optimal control can readily be extended to quantum systems (10) and has become a field of growing interest (11–13).

Although the main source of examples presented is liquid‐state nuclear magnetic resonance (NMR), the techniques shown here are in no way confined to ensemble quantum computing but hold for single‐spin solid‐state quantum computing (14), electron spin resonance (ESR), and techniques beyond spin dynamics such as charge or flux qubits in a Josephson element (15). Methods of geometric control on Lie groups (16) apply to all quantum systems whose dynamics are governed by finite‐dimensional Lie algebras within the framework of spin or pseudospin systems (at least to sufficient approximation).

Quantum computational qubit systems may be implemented with particular convenience by nuclear spins‐1/2, since spin degrees of freedom are largely isolated from their environment. Moreover, the isotropic overall tumbling of the molecules in a liquid sample decouples the, say, spins within each molecule from all their surrounding ones, and the spins can readily be represented by a density matrix. It carries the ensemble average over all molecules, each with spins (4). Thus, the spin degrees of freedom span a Hilbert space of dimension as desired in systems of qubits.

It is the ease of experimental control and theoretical setting that gave NMR a head start in the experimental realization of fundamental concepts in quantum computing (17–19).

As a liquid NMR sample contains an ensemble of many spin systems of the same kind, one can neither manipulate nor detect individual ones, thus precluding the preparation of pure states. Nevertheless, in order to use the usual quantum algorithms designed for initial conditions in terms of pure states, one may transcribe the density operator of a spin ensemble to a so‐called pseudopure state (18). Furthermore, ensemble‐averaged expectation values are detected rather than observables of individual spin systems. Hence, NMR quantum processors are examples of expectation‐value quantum computers (EVQC), where the outcome of a given quantum algorithm can be extracted from the resulting NMR spectra. Under mild conditions, spin‐1/2 systems are fully controllable in the sense that every unitary transform can be realized experimentally (vide infra), and thus universal sets of elementary quantum gates (20) can be put into practice. Then the basic steps of quantum algorithms can be implemented by spin‐selective radio‐frequency pulses (see Figure 28.1) and, for example, CNOT (controlled NOT) gates. In this manner, many algorithms were carried out with NMR experiments, such as the Deutsch–Jozsa algorithm for two (21,22), three (23), and five (24) qubits; variations of Grover's algorithm for two (25,26) and three qubits (27); the period‐finding algorithm for five qubits (28); and a pioneering version of Shor's algorithm for seven qubits (29). So NMR emulation of quantum computing has also been extensively reviewed, for example, in ( 3 30–42); in turn, for the specific importance of optimal control in NMR, see also the reviews ( 3,43).

28.2 From Controllable Spin Systems to Suitable Molecules

28.2.2 Molecular Hardware for Quantum Computation

From a chemical perspective, compounds with suitable spin systems require molecules with coupled spins‐1/2. To this end, not all of the spins have to be mutually coupled, but they have to form a connected coupling topology, so there should be no working qubits without any coupling to the other ones. In particular for large qubit systems, linear spin chains with coupling topologies of nearest‐neighbor interactions ( ) are far more realistic than complete coupling topologies ( ) as shown in Figure 28.2. If controlled individually, arbitrary coupling terms can be used, such as combinations of isotropic and dipolar couplings.

In order to perform a large number of basic computational steps in a quantum algorithm, the time for each quantum gate must be considerably smaller than the relaxation time of the qubits. Moreover, it is highly desirable to strive for time‐optimal implementations of quantum algorithms or their modules in order to avoid unnecessary decoherence. The particular strength of optimal control for achieving this goal will be shown in Section 28.5.

Currently used sample preparations for liquid‐state NMR quantum computers result in nuclear spin relaxation times of up to several seconds. Characteristic spin–spin coupling constants are of the order of 10– Hz, resulting in a typical duration of two‐qubit quantum gates between directly coupled spins of s. Hence, sequences of up to – two‐qubit quantum gates are feasible based on current liquid‐state NMR technology, and even more quantum gates may be possible by increasing the spin–spin coupling constants, for example, by using dipolar couplings in liquid crystalline media (54) and by further increasing the relaxation times. Compared to two‐qubit operations, single‐spin quantum gates such as NOT or Hadamard gates are very short. For example, in heteronuclear spin systems, typical single‐spin gate durations are of the order of s. The minimum time required for a given single‐spin quantum gate not only depends on the maximum amplitude of radio‐frequency pulses but also on the smallest frequency difference of the nuclear spins in a given molecule (24).

For the first NMR quantum computers with up to three qubits, readily available compounds were used, such as 2,3‐dibromothiophene ( 17,55), C‐chloroform (22), 2,3‐dibromopropanoic acid ( 22,56), and ‐alanine (57). For the realization of the first five‐qubit NMR quantum computer, the compound BOC‐( ‐ N‐ ‐Gly)‐F was synthesized ( 24,58) (see Figure 28.3). If the deuterium spins are decoupled, the nuclear spins of , N, , (i.e., ), and F form a coupled spin system consisting of five spins‐1/2. The coupling constants range between 13.5 Hz ( ) and 366 Hz ( ), and for a magnetic field of 9.4 T, the smallest frequency differences are 12 kHz ( ). Further synthetic five‐qubit systems are (i) a perfluorobutadienyl iron complex as entirely homonuclear spin system consisting of five coupled F spins (28) and (ii) ‐ N‐diethyl‐(dimethylcarbamoyl)fluoromethylphosphonate as fully heteronuclear spin system allowing for fast selective pulses (59). A carbon‐labeled analog to (i) has been used as a seven‐qubit molecule for implementing a variant of Shor's algorithm (29). Another seven‐qubit molecule suggested for NMR quantum computing applications is ‐crotonic acid (60). The design and synthesis of molecules with suitable spin systems for 10–20 qubits is not a trivial chemical challenge. An alternative way of realizing a molecular architecture with more than 10 coupled spins is the synthesis of polymers with a repetitive unit consisting of three or more spins (53). This approach is appealing because only a small number of resonances have to be addressed selectively. However, in such an architecture the implementation of quantum algorithms will require an additional overhead, which poses new challenges for the efficient implementation of quantum gates.

28.3 Scalability

28.3.1 Scaling Problem with Pseudopure States

The density operator of a spin system at thermal equilibrium is proportional to , where is the spin Hamiltonian of the ‐spin molecule used as a quantum register, is Boltzmann's constant, and is the temperature. As the usual magnetic fields in NMR are of the order of 10 T, the so‐called high‐temperature approximation is valid above temperatures of some 10 mK, and the thermal density operator can be given by the first two terms in the Taylor expansion

28.1

where is the angular frequency of the th nucleus and is defined by a tensor product over all spins in which all the factors are unit operators except for as the th factor of the tensor product.

Although highly mixed, this state can be transformed into a so‐called pseudopure state, ( 17, 18,61) resulting in an initial density operator of the form

28.2

for some (usually small) coefficient . With the identity operator being invariant under any similarity transform and all spin observables being traceless, pseudopure states form handy starting points for NMR implementations of quantum algorithms. However, this convenience comes at a high cost: the coefficient decreases exponentially with the number of qubits (62). Hence, the spectroscopic signal decreases as well, and severe signal‐to‐noise problems are expected for experiments with more than about 10 qubits.

The exponential signal loss is often thought to impose a fundamental limit on the scalability of ensemble NMR (63), although hyperpolarization techniques (64–66) come very close to pure states. Fortunately, the purity problem can also be circumvented by several approaches avoiding pseudopure states altogether.

28.3.2 Scalable Quantum Computing on Thermal Ensembles

One attractive approach avoiding pure or pseudopure states altogether is to design ensemble quantum computing algorithms based on the thermal density operator instead of a pure state, an early example being a scalable version of the Deutsch–Jozsa algorithm (67,68). At the expense of an extra qubit and a modified oracle, balanced functions can be distinguished from constant ones using an initial state obtained merely by a hard ‐pulse applied to the thermal state. This requires neither pseudopure states of Eq. 28.2 nor temporal averaging. Let denote the number of levels in an ‐qubit system. Then, for an Oracle of a function , one implements a substitute for instead of . To this end, relate to by

Given the Oracle , a scalable NMR quantum computer can readily discriminate balanced functions from constant ones. Note that resolving the output spectra (67) does not build upon any demands growing exponentially with the number of qubits. For example, for the constant function and the balanced function , the scalable version of the Deutsch–Jozsa algorithm requires an additional qubit ( ) and the implementation of for and of . For the five‐qubit system BOC‐( ‐ N‐ ‐Gly)‐F ( 24, 58), the resulting spectra (68) of are shown in Figure 28.4a,b for and , respectively. Constant and balanced functions can be easily distinguished by the presence or absence of the signals.

It is important to note that for this version of the algorithm, the number of molecules in the ensemble does not have to increase exponentially with the number of qubits within the molecule. These favorable scaling properties are at variance to a previous alternative ensemble implementation of the Deutsch–Jozsa algorithm (69,70).

A more recent approach is to perform quantum algorithms designed to work on ensembles as addressed in the worked example of Section 28.6: It is DQC1‐type algorithm to classify knots by topological invariants inferred from measuring spin ensembles.

However, with an increasing number of qubits, not only the synthetic requirements grow, but also the control demands with respect to NMR instruments and pulse‐sequence design, in order to cope or circumvent experimental imperfections (such as rf inhomogeneity) or relaxation. This asks for optimal control methods (3) at large.

28.4 Algorithmic Platform for Quantum Control Systems

In practice, quantum control problems amount to steering a dynamic system such as to maximize a given figure of merit subject to the constraint of following a given equation of motion. In (finite‐dimensional) quantum dynamics, the pertinent equations of motion are typically linear both in the state as well as in the control terms, and dynamic systems of this form are known as bilinear control systems ( 9,71,72)

with “state” , drift , controls , and control amplitudes thus defining the as effective generators. This setting captures all of the following important scenarios:

1. controlled Schrödinger equation⁰¹
   
2. quantum gate for closed system
   
3. quantum state in open quantum system
   
4. quantum map of open quantum system
   

Moreover, the quality function may be expressed via the scalar product as the overlap between the final state (or operator) of the controlled system at time and the target state so that the common options amount to

Define the boundary conditions as , and fix the total time . For simplicity, we henceforth assume equal discretized time spacing for all timeslices . So . Then, the total generator (i.e., Hamiltonian or Lindbladian ) governing the evolution in the time interval shall be labeled by its final time as

which governs the controlled time evolution in the timeslice . Then, gradient‐based and second‐order optimal control algorithms such as GRAPE (73–75) or KROTOV type ( 11 76–79) proceed in the following basic steps entering the unified modular platform DYNAMO (74) described in more detail below.

1. initialize with a random (or guessed) control vector (pulse sequence) consisting of the piecewise‐constant control amplitudes ;
2. exponentiate for all with ;
3. calculate forward‐propagation ¹ ;
4. calculate back‐propagation
5. evaluate fidelity say , where ;
6. evaluate gradients for all : with of Eq. 28.3 or 28.4 below and ;
7. update amplitudes for all , for example, by for a quasi‐Newton second‐order increment, where is a suitable step size and is (e.g., an LBFGS‐approximation to) the inverse Hessian;
8. reiterate up to terminal condition (e.g., for all ).

Here, the exact derivative in closed systems (or unital open systems characterized by their normal Lindblad generators) can be read element‐wise from the eigendecomposition (with eigenvectors to the eigenvalues )

28.3

while in nonunital open systems other methods apply like

28.4

as long as the digitization by is sufficient to satisfy , or one will have to resort to finite‐differences, and so on (see ( 73– 75)). This scheme covers all the optimal control problems specified earlier.

Recently, we have provided a unified MATLAB‐based programming frame DYNAMO (74) designed in a modular way such that to the above set of bilinear control problems it embraces the different algorithmic approaches known in the literature and shown in Figure 28.5. While the GRAPE algorithm (gradient‐assisted pulse engineering) (73) updates all timeslices in the pulse sequence (control vector) concurrently, another type of well‐established algorithms of KROTOV type ( 11 76– 79) do so sequentially. It has turned out that for optimizing unitary gate synthesis for quantum information, concurrent updates of GRAPE type overtake sequential algorithms of KROTOV type well before reaching qualities in the order of the error‐correction threshold, while for spectroscopy purposes lower fidelities that KROTOV may reach faster often suffice. This is due to the fact that the recursive scheme (BFGS) to approximate the inverse Hessian pays when a constant set of time slices is updated as in GRAPE, while sequential updates preclude full profit from such recursions for second‐order methods, and their first‐order variants naturally loose power in the vicinity of critical points. In DYNAMO, one may easily change between different schemes on the fly during an optimization run, whenever needed to save computation time. Moreover, DYNAMO can readily be kept state of the art with respect to future developments such as, for example, improved preconditioning, further Newton‐type algorithms, or including incoherent degrees of freedom as control parameters.

28.5 Applied Quantum Control

Although one can decompose any quantum computing algorithm into a series of single‐spin operations and two‐spin gates between directly coupled spins, some fundamental questions remain: they are of both theoretical and practical interest. What is the minimum time required to realize a given unitary transformation in a given coupling topology of a spin system with a required fidelity? Which controls (pulse sequences) achieve the task in minimal time?

In addition to numerical approaches ( 73, 74), where by repeating controlled state transfer with decreasing final times up to a minimal time still allowing to get full coherence transfer (see ( 10–12, 73, 80)) optimal control theory has also provided analytical approaches and solutions for time‐optimal quantum transfers and gates (81–91).

Moreover, one may characterize time‐optimal pulse sequences algebraically by geometric optimal control (16) showing that the problem reduces to finding geodesics (i.e., shortest paths) between cosets (81), as will be demonstrated in Section 28.5.1.3.

But before that we focus on quantum gate control ( 10, 73 92–94).

28.5.1 Regime of Fast Local Controls: the NMR Limit

Firstly, we choose the limit of fast local controls (by strong pulses), the timescale of which can safely be neglected as compared to the time‐limiting coupling interactions of the Ising type. Not only is this regime typical of NMR with weak scalar couplings, it also lends itself for a theoretical understanding in Lie‐algebraic terms. Here, the ‐couplings are assumed to be uniform in the following examples, thus allowing to express the time required in units of . However, the numerical algorithms are of course general and can cope with coupling types and strengths directly matching the experimental settings, and even finite durations for local controls can be dealt with as shown in Section 28.5.2.

28.5.1.1 The Quantum Fourier Transform

The quantum Fourier transform (QFT) is in the core of all quantum algorithms of Abelian hidden subgroup type (95,96) such as, for example, the algorithms of Deutsch–Jozsa's, Simon's, and Shor's. In order to speed up quantum modules and minimize decoherence, the QFT should be implemented in the fastest way. Clearly, the time required for realizing the QFT in ‐qubit systems depends on the coupling topology and the interaction type and strength of the pertinent experimental setting. Figure 28.6 demonstrates how in linear spin chains ( ) with the nearest‐neighbor Ising interactions, numerical time‐optimal control provides a decomposition of the QFT that is much faster than the corresponding decomposition into standard gates would impose: in six qubits, for instance, the speedup is more than eightfold and in seven qubits approximately ninefold.

28.5.1.2 Multiply Controlled NOTs

Analogously the NOT‐gate can be decomposed in a time‐optimized way. Interestingly, in a complete coupling topology of qubits, the algorithmic complexity was described by Barenco et al. (99) as growing exponentially up to six qubits, whereas the increase from seven qubits onward was said to be quadratic. Again, time‐optimal control provides a dramatic speedup in this case, see Figure 28.7.

28.5.1.3 Geometry of Time‐Optimal Gates

In NMR, the markedly different timescales for fast local controls (pulses) versus slow coupling evolutions lend themselves for making use of Cartan decomposition of real semisimple Lie algebras (where ). The goal of time‐optimal realizations then reduces to finding constrained shortest paths in the cosets . For spins‐1/2, , , and the coset takes the form of a Riemannian symmetric space. Thus, time‐optimal trajectories between points in correspond to Riemannian geodesics. For , the cosets are no longer Riemannian symmetric spaces, so the time‐optimal trajectories in denote sub‐Riemannian geodesics.

Yet, in the sub‐Riemannian geometry of three spins, there are examples that can be fully understood: for example, the time‐optimal simulation of three‐spin‐interaction Hamiltonians of the form where , , can be , or and is an effective trilinear coupling constant. We considered a linear coupling topology consisting of a chain of three heteronuclear spins‐1/2 with the coupling constants , and the coupling term Here, the time‐optimal realization of the trilinear coupling term , see Figure 28.8, can be derived by means of geometric control (100).

Compared to conventional approaches (101), the time‐optimal synthesis of effective three‐spin propagators of the form has a duration of only (100) compared to the duration of conventional implementations (101) and hence provides significant time savings as shown in Figure 28.9. Further results on geometric control can be found in ( 81– 91).

28.6 Worked Example: Unitary Controls for Classifying Knots by NMR

Many of the well‐established quantum algorithms operate by solving the hidden subgroup problem in an efficient way (104,105). Moreover they do so by resorting to the circuit model with its experimentally challenging accuracy demands (error‐correction threshold). In search for different and more robust classes of quantum algorithms, topological quantum computing with anyonic quasi‐particles brought up relations to braid groups (106–108). This is because anyonic world lines in a three‐dimensional model of spacetime (comprising two spatial and one temporal dimension) form braids that can be exploited as quantum gates. These gates have the power of the circuit model with the advantage of being more robust. When establishing the relation between topological and ordinary quantum computation, it turned out that unitary representations of braid groups useful for anyonic topological quantum computing can also be used to compute invariants of knots and links such as the Jones polynomial.

Thus, there is a fruitful interplay between topological and circuit‐based algorithms mediated via braid groups of knots, that is, by unitary representations of the braid operations. In order to implement these unitaries experimentally, optimal control is pertinent again.

Resuming (109,110), in this section we illustrate how thermal ensembles can be used for approximating the trace of a unitary matrix (68) in order to classify knots by their topologies. This paves the way to a recent class of quantum algorithms related to knot theory, because it allows for efficiently evaluating Jones polynomials over a range of parameters. Since knots with different Jones polynomials are clearly inequivalent (while the converse does not hold), efficient quantum algorithms determining the trace of unitaries can be of great help (in the cases distinguishable by the Jones polynomials) to solve the classically NP‐hard decision problem whether two knots are equivalent in the sense they can be transformed into one another by using only Reidemeister moves and trivial moves, that is, those which do not change the number of crossings.

More precisely, while a knot is defined as an embedding of the circle in three‐space up to ambient isotopy, a link is an analogous embedding of several disjoint circles again up to isotopy. Now a knot invariant is any function that remains invariant under Reidemeister (and trivial) moves mentioned already. The Jones polynomial is a special form of Laurent polynomial (i.e., a polynomial with terms of both positive and negative degrees forming a ring) which itself has a degree that grows at most linearly with the number of crossings in the corresponding link. Note there is an important relation to braid groups established by Alexander's theorem. It says that any link can be constructed as a plat closure of some braid, namely by moving “return” strands back into the braid, see, for example, Ref. (111) for details.

Now the algorithm of Aharonov et al. (107,112,113) takes the trace of some unitary representation of the corresponding braid group to give the Jones polynomial. Here the braid group with strands, , is generated by its generators representing right‐handed twists . For evaluating the trace, it is most convenient to exploit the connection to the Temperley‐Lieb algebra (114,115) and its unitary representation by

where is of modulus 1 and is real symmetric, while is the generator of the braid group associated to the knot of interest.⁰²

Next, we focus on the three‐stranded braid group generated by the elements . It comprises the well‐known standard knots Trefoil (up to addition of a circle disjoint from the knot), Figure‐Eight, and the Borromean Rings shown in Figure 28.10.

In the unitary (path model) representation of one ends up with the following unitaries that contain (related to the variable of the bracket and Jones polynomial).

Now, in order to get hold of the trace of by a quantum measurement, we follow Ref. (68) and enlarge the quantum register by one ancilla qubit. Then, the unitary is translated into a controlled unitary with respect to the ancilla in the sense

Based on the thermal ensemble state with , it is routine (here on the molecule chloroform by H saturation followed by gradient filters) to prepare the suitable initial state with the ‐magnetization on the natural abundance C used as qubit. With these stipulations it is easy to proceed in three final steps

1. prepare .
2. let evolve under the signature sequence (vide infra) of 's specific to the knot in question. This gives the total .
3. measure the expectation value of the phase‐sensitive ensemble detection operator⁰³ as to give .

In simple cases it is well known how to translate unitary operators into NMR pulse sequences. In the more intricate case here, similar recipes apply: by combining algebraic approaches with numerical ones (e.g., by GRAPE), one arrives at the pulse sequences shown in Figure 28.11, which are specifically designed to continuously depend on the variable via

so that they can be implemented over a range of values of .

Now, for the Trefoil knot the NMR pulse sequence for has to be applied thrice , while for the Figure‐Eight knot it is and for the Borromean Rings to be read from right to left to give the respective and . As shown in Figure 28.12, the Jones polynomial was experimentally evaluated for each knot or link at 31 values of distributed over a continuous part of the domain accessible by the quantum algorithm. This approach (110) readily discriminates the three‐stranded knots or links by two qubits, while in Ref. (116) only single values of were used. Note that the experimental data nicely follow the theoretical prediction, and the functional dependence is so different that the predictive power of distinguishing knots or links is higher than by mere evaluation of single points.

Yet both experimental demonstrations include an evaluation of the Jones polynomial at a root of unity and thus implement a DQC1‐complete quantum algorithm (see (117)). In Ref. (116), only links that contain disjoint circles were evaluated. As already mentioned, a much simpler quantum calculation using fewer qubits (here two qubits for a two‐strand braid representation) can calculate the Jones polynomials of the given links equally well. In contrast, the evaluations for the Figure‐Eight knot and the Borromean rings cannot do with fewer than three strands and two qubits as shown in Ref. (109).

Even moderately intricate molecular hardware with several qubits and realistic coupling topologies goes beyond pulse sequences as easy as in Fig. 28.11 for the two‐qubit molecule chloroform. Already the four‐carbon architecture used in (116) required the GRAPE algorithm to be implemented experimentally. Hence, control algorithms will play a role for future algorithms inspired by topological quantum computation.

28.7 Conclusions

Apart from low temperatures, hyperpolarization techniques, or in situ reactions with para‐hydrogen (where ensemble states of high purity can be obtained), even thermal ensemble states may be used for NMR implementations of quantum algorithms, which are in principle scalable (118). Beyond the early example of an ensemble variant to the Deutsch–Jozsa algorithm (63), we discussed a DQC1 quantum algorithm ( 109, 116, 117) to classify knots by their topological invariants (Jones polynomials) easily read from spin ensembles.

Acknowledgments

This work was supported in part by the integrated EU‐programme SIQS as well as by DFG (Deutsche Forschungsgemeinschaft) in the collaborative research centre SFB 631 on solid‐state‐based quantum computation.

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