aa r X i v : . [ qu a n t - ph ] J a n Chapter 1
Gateway schemes of quantum control for spinnetworks ⋆ Koji Maruyama and Daniel Burgarth Department of Chemistry and Materials Science, Osaka City University, Osaka558-8585, Japan Institute of Mathematics and Physics, Aberystwyth University, Aberystwyth SY233BZ, UK
Towards the full-fledged quantum computing, what do we need? Obviously, the firstthing we need is a (many-body) quantum system, which is reasonably isolated fromits environment in order to reduce the unwanted effect of noise, and the secondmight be a good technique to fully control it. Although we would also need a well-designed quantum code for information processing for fault-tolerant computation,from a physical point of view, the primary requisites are a system and a full controlfor it. Designing and fabricating a controllable quantum system is a hard work in thefirst place, however, we shall focus on the subsequent steps that cannot be skippedand are highly nontrivial.Typically, when attempting to control a many-body quantum system, every sub-system of it has to be a subject of accurate and individual access to apply operationsand to perform measurements. Such a (near-) full accessibility leads to a problem ofnot only technical difficulties, but also noise (decoherence), as the system can read-ily interact with its surrounding environment. In a sense, we are wishing for twoinconsistent demands, namely, being able to manipulate a quantum system fully bycontrolling the field parameters while suppressing its interaction with the field. ⋆ Cite as: K. Maruyama and D. Burgarth,
Gateway schemes of quantum control for spin networks ,Chapter 6 in T. Takui, L. J. Berliner, and G. Hanson (eds.), Electron Spin Resonance (ESR) BasedQuantum Computing, Springer New York, pp 167–192 (2016). 1 Koji Maruyama and Daniel Burgarth A good news is that the technological progress over the last decades has beenso great that we are now able to access and control quantum systems quite well,provided they are not too large. The coherent manipulations of small quantumsystems, in addition to the observations of quantum behaviours, have been repor-ted for various systems, e.g., NMR/ESR [1, 2, 3, 4], semiconductor quantum dots[5, 6, 7], superconducting quantum bits (qubits) [8, 9, 10], and NV-centres in dia-monds [11, 12].Here, we discuss a possible scheme to bridge the gap between what we wish toachieve and what we can realise today. Namely, we aim at controlling a given many-body quantum system and identifying it by accessing only a small subsystem, i.e., gateway . Restricting the size of accessible gateway and minimising the number ofcontrol parameters should be of help in suppressing the effects of noise.This chapter consists of two parts, each of which is devoted to these two topics,full quantum control through a gateway and Hamiltonian identification, respectively.Such situations, in which only a subsystem is accessible, arise for example in net-works of ‘dark spins’ in diamond and solid state quantum devices[12, 13, 14] aswell as spin networks in NMR and ESR setups [1, 4, 15].In the first part, we present how a system can be controlled through access to asmall gateway. Starting with a general argument on the controllability of a quantumsystem, we show a possible scheme to control spin networks under limited access.The two major issues of our interest in terms of the controllability concern the algeb-raic criterion for the form of Hamiltonians and the topological (or graph theoretical)condition for the choice of gateway. While the consideration about these aspectswill lead to clear insights into the control of spin-1/2 systems, the theory is generalenough to be applied to other systems we encounter in the lab. We shall also discussa few issues related to efficiency, such as, can we compute a pulse sequence for acertain unitary on the chain by a classical computer within polynomial time? Orhow much time would a unitary require to be performed?All these discussions on the controllability assume the complete knowledge ofthe system Hamiltonian. The second part of this chapter is devoted to the discus-sions on how the Hamiltonian can be identified despite the limited access. Withoutthe knowlege of Hamiltonian, we can never control a quantum system at will: it willbe like going for treasure hunting without a map and a compass. Having learned thedetails of the system Hamiltonian, we then attempt to fully control it, enjoying thequantumness of the dynamics. Nonetheless, both the full information acquisitionand the full control are still very hard. In addition, the operational complexity ofinformation acquisition (state and process tomographies) grows rapidly (exponen-tially) with respect to the system size.Presumably the most straightforward way to estimate the quantum dynamics isto apply quantum process tomography (QPT), which is a method to determine acompletely positive map E on quantum states. The map E on a state ρ can be writtenas E ( ρ ) = P i E i ρE † i , where the operators E i satisfy P i E † i E i = I (if E occurswith unit probability) [16]. The complexity of QPT grows exponentially with respectto the system size; for a N qubit system, we need to specify N parameters for E and it is an overwhelming task even for small qubit systems [17, 18, 19]. Moreover, Gateway schemes of quantum control for spin networks 3
QPT necessitates estimating all the matrix elements of ρ , the state of the wholesystem, which is impossible under a restricted access with zero or little knowledgeon the Hamiltonian.The hardness of the task stems from our complete ignorance about the natureof the dynamics. However, here we will consider the cases in which some a prioriknowledge or good plausible assumptions are available to us. In reality, it is naturalto have substantial knowledge on a fabricated physical system, which is the subjectof our control, due to the underlying physics we intend to exploit. Thus, here we willsee how such a priori information on the system can help reduce the complexity ofHamiltonian identification. We will primarily focus on the systems consisting ofspin-1/2 particles. This is largely because they have been attracting much attentionrecently as a promising candidate for the implementation of quantum computers.Yet, it would not make much sense if the size of the gateway is comparable tothat of the entire system. From the viewpoint of noise suppression, the smaller thegateway size, the better. Then how can we find a minimal gateway that suffices toobtain full knowledge on the system? As we will see below, the same graph propertywe introduce in the first part, i.e., the study of spin network control, comes in to thediscussion as a criterion for estimability of the spin network Hamiltonian.This Chapter is based on the results from [20, 21, 22, 23, 24] as well as somenew results. art I Indirect control of spin networks
A central question in control theory is provided a system, typically described bystates, interactions, and our influence on them, to characterize the operations thatcan be achieved by suitable controls. In (unitary) quantum dynamics, the usual setupis a time dependent Hamiltonian of the form H ( t ) = H + X k f k ( t ) H k , (1.1)where the time dependence f k ( t ) can be chosen by the experimentator. While inusual quantum mechanics we solve the Schrödinger equation for a given f k ( t ) toobtain a time evolution unitary U, the question of control is exactly the inverse:provided a unitary U, is there a control f k ( t ) which achieves it? The unitaries forwhich this is true are called reachable. Given a system (1.1), how do we characterize the reachable unitaries? It turnsout that it is easier to include those unitaries which are reachable arbitrarily well into our consideration, and to describe things in terms of simulable Hamiltonians: we call a Hamiltonian iH simulable if exp( − iHt ) is reachable arbitrarily well forany t ≥ . Clearly, iH is effectively reachable by setting f k ≡ and letting thesystem evolve for a suitable time t. We could also set f ≡ and all others zero,and simulate iH + iH , and so on. Let us call the simulable set L and see whichrules it obeys:1. A, B ∈ L ⇒ A + B ∈ L : this is a simple consequence of Trotter’s formula,which says that by switching quickly between A and B the system evolves underthe average of A and B. A ∈ L , α > ⇒ αA ∈ L : this follows simply from letting a weaker interactionevolve longer to simulate a stronger one, and vice versa.3. A, − A, B, − B ∈ L ⇒ [ A, B ] ∈ L : this follows from a not so well-knownvariant of Trotter’s formula given by lim n →∞ (cid:16) e Bt/n e At/n e − Bt/n e − At/n (cid:17) n = e − [ A,B ] t (1.2)4. A ∈ L ⇒ − A ∈ L : This is a property which heavily relies on finite dimensions,where the quantum recurrence theorem holds, ∀ ǫ, t > ∃ T > t : || e − AT − || ≤ ǫ (1.3)which implies e − A ( T − t ) ≈ e + At . If we combine all the above properties we find that the simulable set obeys ex-actly the properties of a Lie algebra over the reals. This is very useful; in particular,if through rules 1-4 arbitrary
Hamiltonians can be simulated, then likewise arbit-rary unitaries are reachable: the system is fully controllable [25, 26, 27] (in fact,this condition is necessary and sufficient) . It was shown by Lloyd that it is a generic property: in fact two randomly chosen Hamiltonians are universal for quantum com-puting almost surely. We will not prove this here as we are going to show somethingstronger: a randomly chosen pair of two-body qubit Hamiltonians is universal forquantum computing almost surely. That is, Lloyd’s result holds even when restrict-ing ourselves to physical
Hamiltonians.
The above equations do not yet take into account the structure of the controls. As dis-cussed in the introduction, it is interesting to consider the case of composite system V = C S C where only a part C of the system is controlled, while the remainder C is completely untouched. In the light of Eq. (1.1) this means that H k = h ( k ) C ⊗ C . Control is mediated to C only through the drift H = H V , which acts on C and C . If through H V the whole system is controllable, it means that we have a caseof weak controllability: the controls H k do not themselves generate all Hamiltoni-ans, the drift evolution is necessary. This implies that H V sets a time limit for howquickly the system can be controlled. It also reveals many-body properties of H V and is therefore interesting from a fundamental perspective.The question is, given H V and a split of the system into CC, how can we decideif the system is controllable? Is the general result by Lloyd still correct when re-stricting ourselves to such a split, and to a physically realistic H V ? In the following,we will aim to answer both questions.Using the results from the last section, V is controllable if and only if h iH V , L ( C ) i = L ( V ) , (1.4)where, for the sake of simplicity, we have assumed the ih ( k ) C ’s to be generatorsof the local Lie algebra L ( C ) of C and where we use the symbol hA , Bi to rep-resent the algebraic closure of the operator sets A and B . L ( V ) denotes the fullLie algebra of the composite system V. The condition (1.4) can be tested numeric-ally only for relatively small systems. It becomes impractical instead when appliedto large many-body systems where V is a collection of quantum sites (e.g. spins)whose Hamiltonian is described as a summation of two-sites terms. For such con-figurations, a graph theoretical approach is more fruitful. The proposed method exploits the topological properties of the graph defined by thecoupling terms entering the many-body Hamiltonian H V . This allows us to translatethe controllability problem into a simple graph property, infection [28, 29, 30]. Inmany-body quantum mechanics this property has many interesting consequences on the controllability and on relaxation properties of the system [28, 20]. Also, the sameproperty, also called zero-forcing , has been studied in fields of mathematics, e.g.,graph theory, in a different context [31]. Let us start reviewing this infection propertyfor the most general setup, which will show more clearly where the topologicalproperties come from.The infection process can be described as follows. Suppose that a subset C ofnodes of the graph is “infected” with some property. This property then spreads,infecting other nodes, by the following rule: an infected node infects a “healthy”(uninfected) neighbour if and only if it is its unique healthy neighbour. If eventuallyall nodes are infected, the initial set C is called infecting . Figure 1.1 would be helpfulto grasp the picture. (a) (b) (c) (d) Fig. 1.1
An example of graph infection. (a) Initially, three coloured nodes in the region C are‘infected’. As the node l is the only one uninfected node among the neighbours of k, it becomesinfected as in (b). (c) Similarly, l ′ becomes infected by k ′ . (d) Eventually all nodes will be infectedone by one. Note that the choice of C that infects V is not unique. Though we are inter-ested in small C, finding the smallest one is a nontrivial, and indeed hard, problem.Nevertheless, from a pragmatic point of view, the number of nodes we consdier forthe purpose of quantum computing would not be too large to deal with as a graphproblem. The link to quantum mechanics is that each node n of the graph has a quantumdegree of freedom associated with the Hilbert space H n , which describes the n -th site of the many-body system V we wish to control. The coupling Hamiltoniandetermines the edges through H V = X ( n,m ) ∈ E H nm , (1.5)where H nm = H mn are some arbitrary Hermitian operators acting on H n ⊗ H m .Within this context we call the Hamiltonian (1.5) algebraically propagating iff forall n ∈ V and ( n, m ) ∈ E one has, h [ iH nm , L ( n )] , L ( n ) i = L ( n, m ) , (1.6)where for a generic set of nodes P ⊆ V , L ( P ) is the Lie algebra associated with theHilbert space N n ∈ P H n . The graph criterion can then be expressed as follows:Theorem: Assume that the Hamiltonian (1.5) of the composed system V is algeb-raically propagating and that C ⊆ V infects V . Then V is controllable actingon its subset C . Proof: To prove the theorem we have to show that Eq. (1.4) holds, or equivalentlythat L ( V ) ⊆ h iH V , L ( C ) i (the opposite inclusion being always verified). Byinfection there exists an ordered sequence { P k ; k = 1 , , · · · , K } of K subsetsof V C = P ⊆ P ⊆ · · · ⊆ P k ⊆ · · · ⊆ P K = V , (1.7)such that each set is exactly one node larger than the previous one, P k +1 \ P k = { m k } , (1.8)and there exists an n k ∈ P k such that m k is its unique neighbor outside P k : N G ( n k ) ∩ V \ P k = { m k } , (1.9)with N G ( n k ) ≡ { n ∈ V | ( n, n k ) ∈ E } being the set of nodes of V which areconnected to n k through an element of E . The sequence P k provides a naturalstructure on the graph which allows us to treat it almost as a chain. In particular,it gives us an index k over which we will be able to perform inductive proofsshowing that L ( P k ) ⊆ h iH V , L ( C ) i .Basis: by Eq. (1.7) we have L ( P ) = L ( C ) ⊆ h iH V , L ( C ) i . Inductive step: assumethat for some k < K L ( P k ) ⊆ h iH V , L ( C ) i . (1.10)We now consider n k from Eq. (1.9). We have L ( n k ) ⊂ L ( P k ) ⊆ h iH V , L ( C ) i and [ iH n k ,m k , L ( n k )] = [ iH V , L ( n k )] − X m [ iH n k ,m , L ( n k )] , where the sum on the right hand side contains only nodes from P k by Eq. (1.9). It istherefore an element of L ( P k ) . The first term on the right hand side is a commutatorof an element of L ( P k ) and iH V and thus an element of h iH V , L ( C ) i by Eq. (1.10).Therefore [ iH n k ,m k , L ( n k )] ⊆ h iH V , L ( C ) i and by algebraic propagation Eq. (1.6)we have h [ iH n k ,m k , L ( n k )] , L ( n k ) i = L ( n k , m k ) ⊆ h iH V , L ( C ) i . Note that the condition (1.6) is a stronger property than the condition of controlling n, m byacting on n . According to Eq. (1.4) the latter in fact reads h iH nm , L ( n ) i = L ( n, m ) , which isimplied by Eq. (1.6).1 But hL ( P k ) , L ( n k , m k ) i = L ( P k +1 ) by Eq. (1.8) so L ( P k +1 ) ⊆ h iH V , L ( C ) i .Thus by induction L ( P K ) = L ( V ) ⊆ h iH V , L ( C ) i ⊆ L ( V ) . (cid:4) (1.11)The above theorem has split the question of algebraic control into two separateaspects. The first part, the algebraic propagation Eq. (1.6) is a property of the coup-ling that lives on a small Hilbert space H n ⊗H m and can therefore be checked easilynumerically. The second part is a topological property of the (classical) graph. Animportant question arises here if this may be not only a sufficient but also necessarycriterion. As we will see below, there are systems where C does not infect V butthe system is controllable for specific coupling strengths . However the topologicalstability with respect to the choice of coupling strengths is no longer given.An important example of the above theorem are systems of coupled spin- / systems (qubits). We consider the two-body Hamiltonian given by the followingHeisenberg-like coupling, H nm = c nm ( X n X m + Y n Y m + ∆Z n Z m ) , (1.12)where the c nm are arbitrary coupling constants, ∆ is an anisotropy parameter, and X , Y , Z are the standard Pauli matrices. The edges of the graph are those ( n, m ) for which c nm = 0 .To apply our method we have first shown that the Heisenberg interaction isalgebraically propagating. In this case the Lie algebra L ( n ) is associated to thegroup su (2) and it is generated by the operators { iX n , iY n , iZ n } . Similarly thealgebra L ( n, m ) is associated with su (4) and it is generated by the operators { iX n I m , iX n X m , iX n Y m , · · · , iZ n Z m } . The identity (1.6) can thus be verified byobserving that [ X n , H nm ] = Z n Y m − Y n Z m [ Z n , Z n Y m − Y n Z m ] = X n Z m [ Y n , X n Z m ] = Z n Z m [ X n , Z n Z m ] = Y n Z m , where for the sake of simplicity irrelevant constants have been removed. Similarlyusing the cyclicity X → Y → Z → X of the Pauli matrices we get, X n Z m → Y n X m → Z n Y m Z n Z m → X n X m → Y n Y m Y n Z m → Z n X m → X n Y m . Finally, using [ Z n Z m , Z n Y m ] = X m , and cyclicity, we obtain all basis elements of L ( n, m ) concluding the proof.According to our Theorem we can thus conclude that any network of spins coupled through Heisenberg-like interaction is controllable when operating on the subset C ,if the associated graph can be infected. In particular, this shows that Heisenberg-likechains with arbitrary coupling strengths admits controllability when operated at oneend (or, borrowing from [25], that the end of such a chain is a universal quantuminterface for the whole system). Using the graph criterion we found that the dynamical Lie algebra for a Heisenbergspin chain with full local control on the first site H Hsbg + g ( t ) Y + f ( t ) Z (1.13)is su (2 N ) , where H Hsbg is the Hamiltonian describing the Heisenberg-type interac-tion, H Hsbg = P ( n,m ) ∈ E H nm with H nm in Eq. (1.12). We can also see that thealgebra generated by H Hsbg + Y + f ( t ) Z (1.14)is su (2 N ) .Extending further, we can consider the Lie algebra generated by A = H Hsbg + Y and B = Z + 1 . Because X = p ( A, Z ) , where p is a (Lie) polynomial in A and Z , replacing Z with Z + 1 we obtain p ( A, Z + 1) = X + c . Commuting with B we find that Y and therefore also Z and seperately are in the algebra generatedby A and B. This has an interesting implication - namely, that the two Hamiltonians A = H Hsbg + Y and B = Z +1 generate u (2 N ) . These are physical Hamiltonians,because they consist of two-body interactions only. The fact that such pair exists canbe used to prove that almost all pairs of two-body qubit Hamiltonians are universal:to do so, we first observe that we can construct a basis of u (2 N ) through repeatedcommutators and linear combinations of A and B : u (2 N ) = span { p ( A, B ) , . . . , p N ( A, B ) } where the p k are (Lie) polynomials in A and B. The fact that this is a basis can beexpressed equivalently through D ≡ det {| p ) , . . . , | p N ) } 6 = 0 , (1.15)where | p k ) is the vector corresponding to the matrix p k ( A, B ) . Now, parametrizing A and B through A = X n,m,α,β a αβnm σ αn σ βm (1.16) B = X n,m,α,β b αβnm σ αn σ βm (1.17) with σ (0 , , , n ≡ (1 n , X n , Y n , Z n ) we can expand D in Eq. (1.15) as a multinomialin a αβnm and b αβnm . Our result implies that this multinomial is not identical tozero, and therefore its roots have measure zero. Therefore the set of parameters ( a αβnm , b αβnm ) for which the system is not controllable is of measure zero. Butthe parametrization (1.16) holds for arbitrary two-body qubit Hamiltonians, whichconcludes the argument. We note that this argument is easily extended to generalmany-body Hamiltonians. The above results are interesting from the theoretical point of view; however, canthey be practically useful from the quantum computing perspective? The two mainproblems we need to contemplate before attempting to build a large quantum com-puter using quantum control are as follows. First, the precise sequence of actualcontrols (or ‘control pulses’) are generally not computable without already simu-lating the whole dynamics. We need to find an efficient mapping from the quantumalgorithm (usually presented in the gate model) to the control pulse. Secondly, evenif such a mapping can be found, the theory of control tells us nothing about theoverall duration of the control pulses to achieve a given task, and it might take fartoo long to be practically relevant.One approach to circumvent these scaling problems focuses on systems that aresufficiently small, so that we do not already require a quantum computer to checktheir controllability and to design control pulses. In such a case, the theory of timeoptimal control [32] can be used to achieve impressive improvements in terms oftotal time or type of pulses required in comparison with the standard gate model.More complicated desired operations on larger systems are then decomposed (‘com-piled’) into sequences of smaller ones. Yet, the feasibility of this approach is ulti-mately limited by the power of our classical computers, therefore constrained tolow-dimensional many-body systems only.
Fig. 1.2 (color online) Our approach for universal quantum computation works on a chain of N spins. By modulating the magnetic field B ( t ) on qubit , we induce information transfer and swapgates on the chain (red and green lines). The states of the qubits from the uncontrolled register canbe brought to the controlled part. There, the gates from a quantum algorithm are performed bylocal operations. Afterward, the (modified) states are swapped back into their original position.4 The goal of this section is to provide an example where one can efficiently com-pute control pulses for a large system, using the full Hilbert space, and to showthat the duration of the pulses scales efficiently (i.e., polynomially) with the systemsize. We will use a Hamiltonian that can be efficiently diagonalized for large sys-tems through the Jordan-Wigner transformation. A similar scheme was developedindependently in [33]. The control pulses are applied only to the first two spins ofa chain (see Fig. 1.2). The control consists of two parts: one where we will use theJordan-Wigner transformation to efficiently compute and control the informationtransfer through the chain (thus using it as a quantum data bus), and a second partwhere we will use some local gates acting on the chain end to implement two-qubitoperations. To be efficiently computable, these local gates need to be fast with re-spect to the natural dynamics of the chain. Combining the two actions allows us toimplement any unitary operation described in the gate model.More specifically, we consider a chain of N spin- / particles coupled by theHamiltonian H = N − X n =1 c n [(1 + γ ) XX + (1 − γ ) Y Y ] n,n +1 + N X n =1 B n Z n , where X, Y, Z are the Pauli matrices, the c n are generic coupling constants, and the B n represent a magnetic field. Variation of the parameter γ encompasses a widerange of Hamiltonians, including the transverse Ising model ( γ = 1 ; for this casewe require the fields B n = 0 ) and the XX model ( γ = 0 ). We assume that the valueof B can be controlled externally. This control will be used to induce informationtransfer on the chain and realize swap gates between arbitrary spins and the two‘control’ spins , at one chain end. Hence such swap gates are steered indirectly by only acting on the first qubit.In order to focus on the main idea we now present our method for γ = 0 and B n = 0 for n > . The general case follows along the same lines, thoughmore technically involved. Our first task is to show that by only tuning B ( t ) , we can perform swap gates between arbitrary pairs of qubits. First we rewrite theHamiltonian using the Jordan-Wigner transformation a n = σ + n Q m 2) = ( | i kl h | + | i kl h | ) ⊗ | i kl h | − | i kl h | ) ⊗ L kl for ( k − l ) even. The operator L kl = Q k All coupling strengths (solid lines) and local magnetic fields (background) of a 2-dimensional network G = ( V, E ) of spins (white circles) can be estimated indirectly by quantumstate tomography on a gateway C (enclosed by the dashed red line). The coupling strengths andfield intensities are represented by the width of lines and the density of the background colour,respectively. This can be answered by using the infecting property, which has been introducedin Sec 1.4 for a given graph G and a subset C ⊂ V of nodes . The main theoremabout hamiltonian identification under a limited access can be presented in terms ofthe infection property as follows. That is, if C infects V, then all c mn and b n can beobtained by acting on C only. Therefore, C can be interpreted as an upper boundon the smallest number of spins we need to access for the purpose of Hamiltoniantomography, i.e., given by the cardinality | C | of the smallest set C that infects V. To prove this statement, let us assume that C infects V and that all eigenvalues E j ( j = 1 , . . . , | V | ) in H are known. Furthermore, assume that for all orthonormaleigenstates | E j i in H the coefficients h n | E j i are known for all n ∈ C. We showhow these information lead to the full Hamiltonian identification, and then in Section1.10 show how these necessary data, E j ( ∀ j ) in H and h n | E j i for all j ∈ , ..., | V | and all n ∈ C , can be obtained by simple state tomography experiments.Observe that the coupling strengths between spins within C are easily obtainedbecause of the relation c mn = h m | H | n i = P E k h m | E k ih E k | n i , where we defined c mm ≡ h m | H | m i for the diagonal terms. Since C infects V there is a k ∈ C and a l ∈ C ≡ V \ C such that l is the only neighbour of k outside of C, i.e. h n | H | k i = 0 ∀ n ∈ C \{ l } . (1.19)For an example see Fig. 1.1. Using the eigenequation, we obtain for all jE j | E j i = H | E j i = X m ∈ C h m | E j i H | m i + X n ∈ V \ C h n | E j i H | n i . Multiplying with h k | and using Eq. (1.19) we obtain E j h k | E j i − X m ∈ C c km h m | E j i = c kl h l | E j i . (1.20)By assumption, the left-hand side (LHS) is known for all j. This means that up toan unknown constant c kl the expansion of | l i in the basis | E j i is known. Throughnormalisation of | l i we then obtain c kl , thus c kl (by using the assumed knowledgeon its sign) and hence h l | E j i . Redefining C ⇒ C ∪ { k } , it follows by induction thatall c mn are known. Finally, we have c mm = h m | H | m i = E − ∆ X n ∈ N ( m ) c mn + 2 b m , (1.21)where N ( m ) stands for the (directly connected) neighbourhood of m, and E = 12 ∆ X ( m,n ) ∈ V c mn − X n ∈ V b n (1.22)is the energy of the ground state | i . Summing Eq. (1.21) over all m ∈ V andusing Eq. (1.22), we can have the value of P n ∈ V b n , thus that of E as well, sinceall other parameters are already known. Then we obtain the strength of each localmagnetic field, b m , from Eq. (1.21).An interesting application of the above scheme is a one-dimensional(1D) spinchain with non-nearest neighbour interactions [38]. If spins interact with the next-nearest neighbours in addition to the nearest ones, the whole graph can be infectedby setting the two end spins as C , as shown in Fig.1.4. Similarly, if spins inter-act with up to r -th nearest neighbours, all coupling strengths can be estimated byincluding the r spins at the chain end, from the first to the r -th, in C. Fig. 1.4 An example of graphs for non-nearest neighbour interactions. The graph for next-nearestinteraction (left) can be infected by C as it is easily seen after deforming (right).4 In order to perform the above estimation procedure, we need to know the en-ergy eigenvalues E j in H and the coefficients h n | E j i for all n ∈ C by con-trolling/measuring the spins in C . Suppose the spin 1 is in C. To start, we initialisethe system as | i and apply a fast π/ -pulse on the spin 1 to make √ ( | i + | i ) . This can be done efficiently by acting on the spin 1 only; the basic idea is that bymeasuring the spin 1, and flipping it quickly every time when it was found in | ↑i ,the state of the network becomes | i within a polynomial time with respect to thenetwork size N = | V | . The reason for this is two-fold: the excitation-preservingproperty of the Hamiltonian guarantees that an up-spin cannot be observed morethan N times and the propagation time of up-spins in the network is polynomial in N [39]. Then, we perform quantum state tomography on the spin n ∈ C after atime lapse t . By repeating the preparation and measurements on spin n , we obtainthe following matrix elements of the time evolution operator as a function of t : e iE t h n | U ( t ) | i = X j h n | E j ih E j | i e − i ( E j − E ) t . (1.23)If we take n = 1 and Fourier-transform Eq. (1.23) we can get information on theenergy spectrum in H . Up to an unknown constant E , which turns out to be irrel-evant, we learn the values of all E j from the peak positions. The height of the j -thpeak gives us the value of |h | E j i| for all eigenstates. Thanks to the arbitrarinessof the global phase, we can set h | E j i > . Hence observing the decay/revival ofan excitation at n = 1 we can learn some E j and all the h | E j i .In order to determine h n | E j i for other n ∈ C, we prepare a state at 1 and meas-ure at n. Namely, setting n ( = 1) in Eq. (1.23) allows us to extract the coefficient h n | E j i correctly, including their relative phase with respect to h | E j i . Continuingthis analysis over all sites in C, we get all information necessary for the Hamilto-nian tomography. It could be problematic if there were eigenstates in H that haveno overlap with any n ∈ C , i.e., h n | E j i = 0 . Fortunately, such eigenstates do notexist, as shown in [28]. Therefore we can conclude that all eigenvalues in the H can be obtained. Although tomography cannot determine the extra phase shift E , itdoes not affect the estimation procedure (it is straightforward to check that it cancelsout in the above estimation).Note that in order for the information about h n | E j i ( n ∈ C ) to be attained thereshould be no degeneracies in the spectrum of Eq. (1.23). For example, suppose thereare two orthogonal states | E (1) k i and | E (2) k i , both of which are the eigenstates of H corresponding to the same eigenvalue E k . The height of the peak at E k in the Fouriertransform of h | U ( t ) | i would be |h | E (1) k i| + |h | E (2) k i| . There is no means toesitmate the value of each term from this sum, let alone the values of h n | E (1) k i and h n | E (2) k i . Also even if there are no degeneracies, thus if E j are all distinct, the peaksneed to be sharp enought to be resolved. The issues on degeneracies and resolvingpeaks are discussed in the following sections 1.11 and 1.11. What if there were degenerate energy levels in the single excitation subspace H ?While 1D spin chains have no degeneracies [40], there could be in general spinnetworks. Of course “exact degeneracy” is highly unlikely; however approximatedegeneracy could make the scheme less efficient. In this section, we show that there always exists an operator B C , which represents extra fields applied on C, such that itlifts all degeneracies of H in H . Because C is only a small subset, the existence ofsuch an operator is not a trivial problem at all. In the following, we demonstrate theexistence of such a B C by explicitly constructing it, assuming the full knowledgeabout H . Without the full knowledge of H (as is the case in the estimation scenario),we could only guess a B C and have it right probabilistically. Nevertheless, as it isclear from the discussion below, the parameter space for B C that does not lift all thedegeneracies has only a finite volume. Thus even choosing B C randomly can makethe probability of lifiting the degeneracies to converge exponentially fast to one.Once all degeneracies are lifted, we can estimate the full Hamiltonian H + λB C ⊗ I ¯ C and subtracting the known part λB C ⊗ I ¯ C completes our identification task. Here, λ is a parameter for the strength of the fields. Although the extra fields on C do notnecessarily have to be a small perturbation, let us consider a small λ to see the effectof λB C on the energy levels, making use of the pertubation theory.Let us denote the eigenvalues of H as E k and the eigenstates as | E dk i , where d = 1 , . . . , D ( k ) is a label for the D ( k ) -fold degenerate states. Let us first look atone specific eigenspace (cid:8) | E dk i , d = 1 , . . . , D ( k ) (cid:9) corresponding to an eigenvalue E k . Since the eigenstates considered here are in H , we can always decomposethem as | E dk i C ¯ C = | φ dk i C ⊗ | i ¯ C + | i C ⊗ | ψ dk i ¯ C , where the unnormalised states | φ dk i C and | ψ dk i ¯ C are in the single excitation sub-space on C and C, respectively. The state | φ dk i C ( ∀ d ) cannot be null, i.e., | φ dk i C =0 , because if there was an eigenstate in the form of | i C ⊗ | ψ dk i ¯ C then applying H repeatedly on it will necessarily introduce an excitation to the region C, in contra-diction to being an eigenstate [28]. Furthermore, the set (cid:8) | φ dk i C , d = 1 , . . . , D ( k ) (cid:9) must be linearly independent: for, if there were complex numbers α kd such that P d α kd | φ dk i C = 0 , then a state in this eigenspace P d α kd | E dk i C ¯ C = P d α kd | i C ⊗| ψ dk i ¯ C would be an eigenstate with no excitation in C, again contradicting the abovestatement. This leads to an interesting observation that the degeneracy of each ei-genspace can be maximally | C |− fold, because there can be only | C | linearly inde-pendent vectors at most in H on C. Thus, the minimal infecting set of a graph, atopological property, is related to some bounds on possible degeneracies, a some-what algebraic property of the Hamiltonian.Now suppose that λ k B kC is a perturbation that we will construct so that it liftsall the degeneracies for an energy eigenvalue E k . Assuming B kC | i C = 0 turns outto be sufficient for our purpose. The energy shifts due to B kC in the first order aregiven as the eigenvalues of the perturbation matrix C ¯ C h E dk | B kC ⊗ I ¯ C | E d ′ k i C ¯ C = C h φ dk | B kC | φ d ′ k i C . We want the shifts to be different from each other to lift the degen- eracy. To this end, recall that (cid:8) | φ dk i ¯ C (cid:9) are linearly independent, which means thatthere is a similarity transform S k (not necessarily unitary, but invertible) such thatthe vectors | χ dk i C ≡ S − k | φ dk i C are orthonormal. The perturbation matrix can thenbe written as C h χ dk | S † k B kC S k | χ d ′ k i C . If we set S † k B kC S k = P d ǫ kd | χ dk i C h χ dk | theHermitian operator B kC ≡ X d ǫ kd (cid:16) S † k (cid:17) − | χ dk i C h χ dk | S − k (1.24)gives us energy shifts ǫ kd . Therefore, as long as we choose mutually different ǫ kd ,the degeneracy in this eigenspace is lifted by B kC . This happens for an arbitrarilysmall perturbation λ k . So we choose λ k such that the lifting is large while no newdegeneracies are created, i.e. || λ k B kC || 6 = ∆E ij , where ∆E ij = E i − E j are theenergy gaps of H. There may be some remaining degerate eigenspaces of the perturbed Hamilto-nian H ′ = H + λ k B kC . Fortunately, since B kC conserves the number of excitations(see Eq. (1.24)), we can still consider only H and repeat the above procedure to findoperators B k ′ C to lift degeneracy in each eigenspace spanned by | E d ′ k ′ i . Eventuallywe can form a total perturbation B C = P k B kC that lifts all degeneracies in H .By perturbation theory a ball of finite volume around B C has the same property. Inpractice, we expect that almost all operators will lift the degeneracy, with a goodcandidate being a simple homogeneous magnetic field on C. This is confirmed bynumerical simulations [23]. The efficiency of the coupling estimation can be studied using standard propertiesof the Fourier transform (see [41] for an introduction). In experiments, the function h n | U ( t ) | m i ( m, n ∈ C ) is sampled for descrete times t k , rather than for continuoustime t , with an interval ∆t = t k +1 − t k . Therefore an important cost parameter is thetotal number of measured points, being proportional to the sampling frequency, f =1 /∆t . The minimal sampling frequency is given by the celebrated Nyquist-Shannonsampling theorem as f min = E max , where E max is the maximal eigenvalue of H in the first excitation sector.Due to decoherence and dissipation, the other important parameter is the totaltime length T (= max( t k )) over which the functions need to be sampled to obtain agood resolution. This is given by the classical uncertainty principle that states thatthe frequency resolution is proportional to /T . Hence the minimal time durationover which we should sample scales as T min = 1 / ( ∆E ) min , where ( ∆E ) min is theminimal gap between the eigenvalues of the Hamiltonian. Also, in order for all peaksin the Fourier transform to be resolved, the height of the peaks, which are given by |h n | E j ih E j | m i| , should be high enough. That is, all energy eigenstates need to bewell delocalised, otherwise most of h E j | m i would have almost zero modulus. Although a coherence time that is as long as T min has been assumed so far tomake the scheme work by letting the signal propagate back and forth many times,the gateway scheme is also applicable to systems with short coherence times bymodifying it. For example, as shown in [42], instead of measuring the spin state inthe accessible area, we may be able to measure in the energy eigenbasis | E n i , andthen the Hamiltonian can be estimated. Such a global measurement is actually easierin some cases than measuring the state of a single component. With this modificationto the scheme, however, the graph condition for the accessible area C needs to beslightly changed; it should be expanded, depending on the graph structure.Another potential concern is the (Anderson) localisation. The localisation of ex-citation (or spin-up) will take place, if there is too much disorder in the couplingstrengths (see, for example, [43]). Then couplings far away from the controlled re-gion C can no longer be probed. In turn, this suggests a way of obtaining inform-ation on localisation lengths indirectly. That we cannot ‘see’ beyond the localisa-tion length would not be a serious problem as our primary purpose is to identify aquantum system we can control.When localisation is negligible, the numerical algorithm to obtain the couplingstrengths from the Fourier transform is very stable [40]. The reason is that the coup-lings are obtained from a linear system of equations, so errors in the quantum-statetomography or effects of noise degrade the estimation only linearly.Let us also look at the scaling of the problem with the number of spins. Typicallythe dispersion relation in one-dimensional systems of length N is cos kN , whichmeans that the minimal energy difference scales as ( ∆E ) min ∼ N − and thus thetotal time interval should be chosen as T min ∼ N . This agrees well with our nu-merical results tested up to N = 100 . For each sampling point a quantum-statetomography of a signal of an average height of N − needs to be performed. Sincethe error of tomography scales inverse proportionally to the square root of the num-ber of measurements, roughly N measurements are required for each tomography. So far, we have focused on the Hamiltonians that preserve the total magnetisation.Nevertheless, it is possible to generalise the above argument to a more general class.They are those that are quadratic in terms of annihilation and creation operators, thatis H = X m,n ∈ E A mn a † m a n + 12 (cid:0) B mn a † m a † n + B ∗ mn a n a m (cid:1) , (1.25)which does not preserve the number of quasi-particles P a † n a n . Here, E is againthe set of interacting nodes as in Eq. (1.18). For H to be Hermitian we must have A = A † and B T = − ǫB, where ǫ = 1 for fermions and ǫ = − for bosons,depending on the particle statistics described by a and a † . For one-dimensional spinchains, the operators a and a † are defined with the standard spin (Pauli) operators through the Jordan-Wigner transformation [44, 45], a † n a n = σ + n Y m A graph corresponding to the matrix M with A and B of Eq. (1.28) and ǫ = 1 (fermionic).For bosonic systems, there will be additional edges connecting nodes m ( ≤ m ≤ N ) and N + m ,because B is symmetric, rather than antisymmetric. For both fermionic and bosonic cases, thereare edges extruding and returning to the same node, corresponding to the diagonal elements of A, which are not shown here to illustrate the principal structure of the graph. Let us take an Ising chain of N spins with transverse magnetic fields, i.e., γ = 1 in Eq. (1.27), as a specific example to demonstrate how the estimation goes. To make use of the symmetry the graph in Fig. 1.5 posesses, let us define | n ± i := 1 √ | n i + | N + n i ) . We already have the information about h ± | E j i , as well as E j , from the measure-ment on the spin 1. The estimation procedure proceeds as in Sec 1.9, namely, bylooking at h + | M | E j i we have E j h + | E j i = − b h − | E j i , whose LHS is known, thus b can be obtained through the normalisation conditionfor h − | E j i . Similarly, evaluating h − | M | E j i gives E j h − | E j i = 2 c h + | E j i − b h + | E j i , from which c and h + | E j i can be known. Also, from E j h + | E j i = 2 c h − | E j i − b h − | E j i we have b and h − | E j i , therefore we have obtained all parameters upto the second spin, so effectively expanded the accessible area to two spins. Then,this procedure can go on one by one till we reach the other end of the chain, i.e., the N -th spin, identifying all the paramters in the matrix M .A remark on the initialisation follows. It was shown in [37] that, in the case of1D XX chains of spins-1/2, the estimation of Hamiltonian parameters is possiblewithout initialising the chain state. The smart trick there was that the spin 1 wasinitialised so that the average value of the z -component of spin, i.e., h Z i , was madezero at t = 0 . The rationale behind it stems from the Jordan-Wigner transform. Since a n = σ + n Q m We have seen that despite a severe restriction on our accessibility a large quantumsystem can be controlled and its Hamiltonian can be identified. As a matter of fact,it is unrealistic for any existing control scheme to have a full access to the system,i.e., a full modulability for the d − parameters for independent Hamiltonians with d being the system dimensionality. In the case of methods based on electron/nuclearspin resonance, for instance, all we modulate is the external magnetic field and wedo not have a full control over all inter-spin couplings. 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