Symmetries on Spin Chains: Limited Controllability and Minimal Controls for Full Controllability
Symmetry is a fundamentally important concept in many branches of physics. In this work, we discuss two types of symmetries, external symmetry and internal symmetry, which appear frequently in controlled quantum spin chains and apply them to study various controllability problems. For spin chains under single local end control when external symmetries exists, we can rigorously prove that the system is controllable in each of the invariant subspaces for both XXZ and XYZ chains, but not for XX or Ising chains. Such results have direct applications in controlling antiferromagnetic Heisenberg chains when the dynamics is naturally confined in the largest excitation subspace. We also address the theoretically important question of minimal control resources to achieve full controllability over the entire spin chain space. In the process we establish a systematic way of evaluating the dynamical Lie algebras and using known symmetries to help identify the dynamical Lie algebra.
Controllability is a fundamental concept in control theory in general, and control of quantum systems in particular. Any quantum system with a sufficient number of controls becomes fully controllable [1, 2, 3, 4]. Therefore we are most interested in the problems where the system has only a limited number of controls and often limited controllability (e.g. subspace controllability). Such limited controllability is usually due to the existence of symmetries in the Hamiltonians [5, 6], which restrict the dynamical Lie algebra (DLA) of the system . Previous literature on quantum controllability has mainly focussed on the cases where either the system is fully controllable (hence implying universal quantum computation ), or not fully controllable but with a DLA that scales linearly or quadratically with the system size. In contrast, in this work, we would like to study systems that are not fully controllable but with a DLA large enough for universal quantum computation.
There are simple criteria for controllability of bilinear systems in terms of the Lie algebra rank condition  similar to the Kalman rank condition for linear systems. However, verifying controllability for quantum systems is challenging, not least because the dimension of the DLA associated with a multi-partite quantum system usually grows exponentially in the number of particles (such as qubits). This exponential scaling makes it impossible in most cases to verify the Lie algebra rank condition numerically. It is therefore important to have general algebraic controllability results for certain classes of systems such as spin chains with a few controls of a certain type. In this paper we derive such results for spin chains with isotropic and even more anisotropic couplings. Unlike spin chains with Ising-type coupling such systems are usually controllable with very few controls acting on a small subset of spins. However, controllability is limited by the existence of symmetries in the Hamiltonians. For instance, it has been shown using the propagation property that Heisenberg chains are fully controllable given two non-commuting control acting on the first spin  but not when there is only single control acting on the first spin. In the latter case the controlled system has symmetries and decomposes into invariant subspaces , preventing full controllability. However, it has been observed that such systems appear to be controllable on each invariant subspace, in particular, the largest excitation subspace, whose dimension scales exponentially with system size .
In this work we give a rigorous proof of this subspace controllability result for XXZ chains and then apply similar techniques to discuss the subspace controllability of a general XYZ spin chain. This system is interesting as it provides arguably the simplest model of a universal quantum computer one could imagine: a physical Hamiltonian with a single control switch to do the computation. We further show that the same result does not hold for XX chains, where a single control acting on the end spin in a chain can only give controllability on a subspace whose dimension does not scale exponentially with system size. In this case additional controls are needed, and we discuss the minimal local control resources for full controllability in this context.
This paper is organized as follows: in Section II, we introduce different types of spin chains and define two fundamental types of symmetry, external and internal, and their relations to controllability. In Section III, we present a complete discussion on spin chains under a single end control, and rigorously prove that for both XXZ and XYZ chains, the system is controllable in each invariant subspace, and that this result is robust if the control field has a leakage on the neighboring spins. In Section IV, we investigate the XXZ or XYZ chains for various types of two controls and we find the minimal control resources for full controllability on the entire Hilbert space. In Section V, we study the dynamical Lie algebra for an XX chain subject to a single end control and investigate the controllability for an XX chain subject to two and three controls.
Ii Model and Basics
For a quantum system composed of spins, we denote the standard Pauli operators by and the local operator acting on the -th spin by , i.e., , where is the identity on a single spin.
System Hamiltonian: We consider a spin network composed of spin- particles with spin-spin interaction characterized by the following Hamiltonian
with the special cases corresponding to isotropic Heisenberg coupling, to XXZ-coupling, and to XX-coupling, and to Ising coupling. For XXZ-networks it is convenient to set and . We also require all couplings in (1) form a connected graph.
The constants determine the coupling strengths between nodes and in the network. Special cases of interest are chains with nearest-neighbor coupling, corresponding, e.g., to linear qubit registers in quantum information processing, for which except when . A network is uniform if all non-zero couplings are equal, i.e., , and . Every spin network has an associated simple graph representation with vertices determined by the spins and edges by non-zeros couplings, i.e., there is an edge connecting nodes and exactly if .
Controllability: The controlled quantum dynamics we are interested in is characterized by the following Schrödinger equation:
where is the system Hamiltonian in (1) and , is a series of control Hamiltonians with time-varying amplitudes . We define the system to be controllable if the dynamical Lie algebra generated by , is equal to the largest Lie algebra or . The definition of controllability is very intuitive: it can be shown that if the system is controllable, then any unitary process can be generated from (2) under certain control sequence in finite time; if , then there exists some unitary gate that can never be generated under (2) . The concepts of controllability and dynamical Lie algebra are very important for both theory and control applications, as they characterize the reachable set of the control dynamics and have answered the question whether a given control task can be achieved or not. However, calculating the dynamical Lie algebra can become extremely difficult or even intractable as increases. Therefore, we hope to use other properties of the Hamiltonians to infer information about controllability, and symmetry does play such a role.
Symmetries: We consider two types of Hamiltonian symmetries: external symmetry and internal symmetry .
Let , , be a set of Hamiltonians for a given quantum system. If there exists a Hermitian operator such that for all then is called an external symmetry for the Hamiltonians; assuming are trace-zero, if there exists a symmetric or antisymmetric operator such that for all , where is the transpose of then is called an internal symmetry.
From the definition, external symmetry implies that all can be simultaneously diagonalized, while internal symmetry implies that the dynamical Lie algebra generated by is a subalgebra of the orthogonal algebra or symplectic algebra . In both symmetry cases, is strictly smaller than and the system is not controllable. It is useful to investigate which operators can be the external symmetries.
Example 1. For the system Hamiltonian (1), a simple class of symmetry operators is of the form , where is a local operator on the -th spin, i.e., . then requires for any connected link in (1), which shows that the nontrivial external symmetry operators are , and , which are often known as the parity symmetry. Hence, if the control Hamiltonians only contain local Pauli operators in one direction, such as direction, with , then is the corresponding parity symmetry and all Hamiltonians are invariant in each of the two eigenspaces of with parity and .
Example 2. If the system Hamiltonian (1) is of XXZ type then defining , we have . Physically, represent the total number of excitations, and has distinct eigenvalues, ranging from to , corresponding to different numbers of excitations in the network. If the control Hamiltonians only contain operators, i.e., then defines an external symmetry, called the excitation symmetry, and all Hamiltonians are block-diagonalized on the invariant subspaces, as illustrated in Fig. 1.
Example 3. A non-identity element in the permutation group defines a permutation symmetry of the spin network if all Hamiltonians are invariant under the permutation of the spin indices . In particular, for a single-local-control problem such as on the -th spin, permutation symmetry means that the index much be fixed under the permutation (Fig. 2). In fact, induces a symmetry operator which commutes with both the system and the control Hamiltonians and hence defines an external symmetry. Moreover, since commute with as defined in Example 2, also induces external symmetries on each excitation subspace of , i.e., all Hamiltonians can be further block-diagonalized in the excitation subspaces.
Having found all external symmetry operators of the Hamiltonians, the entire Hilbert space can be decomposed into , where quantum dynamics is invariant on each , which cannot be further decomposed. The associated dynamical Lie algebra must be a subalgebra of . Although the system is not controllable on the entire space, it may still be controllable on each . In the following, we show that this is indeed true for single local control on the end spin of a XXZ chain, with the symmetry operator as the total excitations.
Iii Single Local End Control
One of the simple but important configurations of a spin network is a spin chain, which is the main subject of the paper. We first consider a spin chain with a single local control at the end of the chain. Without loss of generality, we assume the control field is in Z direction. The corresponding controllability result depends on whether the spin-spin interaction on the other two directions are equal or not, i.e., whether the spin chain is of (1) XXZ type or (2) anisotropic XYZ type.
Iii-a XXZ Chain
For an XXZ chain with spin number , under the end control in Z direction, the system and the control Hamiltonians are written as:
As discussed in previous section, the excitation operator is an external symmetry, and the entire Hilbert space is decomposed into with as the invariant subspace with excitations, i.e., it is generated by the computational basis vectors with number of ’s, where the two single-spin basis vectors are denoted as and . Hence . For example, for , is expanded by , , , , and with . Due to , the controlled system (3) is not fully controllable on the whole space, but it is controllable on each . As an application, for when represents an antiferromagnetic chain, and we can easily prepare the system into the ground state , which is in the largest excitation subspace at . Then, by applying a single control with amplitude derived from optimization, we can generate the total Hamiltonian to drive the system into an arbitrary target state in at a later time . In particular, we can generate perfect entangled pairs between the two end spins of the chain, which is an important quantum resource for many applications such as quantum communication or measurement-based quantum computing .
Next we rigorously prove that under the control dynamics with the Hamiltonians in (3), the system is controllable in each , and particularly in . By definition of controllability, it is sufficient to show that and generate on each . Since , the associated dynamical Lie algebra . The idea of the proof is to determine all independent operators generated in and then evaluate in order to identify .
Since a Lie algebra is also a real vector space, we can drop some factors in the calculation and use linear combinations. We denote such (trivial) steps in the derivation by . First of all, we derive the following commutation relations:
Continuing this process, we can generate all , and (with details in appendix -A). For brevity purposes we will only focus on the terms and not write down the terms explicitly, since one operator can always be generated from the other. An operator is called a -body operator if it contains nontrivial factors, i.e., those comprised of , or Pauli operators. For example, is a 2-body operator, while is a 3-body operator. Denoting as the set of all -body operators in , we list all elements in and evaluate :
() : when is even, we can generate:
when is odd, we can generate:
Next, in order to get , we first evaluate the number of the operators in the form
which contains pairs of or operators. We will call them p-pair operators. For a given and with , we denote the set of p-pair operators as . For example, for , is a 3-pair operator in . Then the size of the set is obtained by simple combinatorics as
However, not all of the elements in are linearly independent. For example, for and , we find
Similarly we can write down the other dependence relations. Altogether there are only of all -pair operators that are linearly independent. In general, we will prove that only of all -pair operators are linearly independent, and
However, directly proving (4) is very difficult as the linear dependence relations can become very complicated for large and . Fortunately, we can convert this problem to evaluating the rank of a set of polynomials on complex field (with details in appendix -B). Therefore, for ,
In the above, the extra combinatorial factor arises from different choices for putting the terms. After some simplification, for both and ,
where the last equation is shown in Lemma 2 in the appendix.
As discussed earlier, , with
All -body -type operators, , generate a Cartan subalgebra in , with . Notice that since we can only generate coupled -body -type operators, such as in , the rank of all -type operators in is , i.e., there are independent -type operators not included in , but included in . We hence have:
It means that achieves the allowed maximal value, which is true only when is isomorphic to on each . Hence, we have proved the following theorem:
For an XXZ chain of length with a single local control on the end spin in Z direction, the system is controllable on each of the invariant excitation subspaces.
In particular, this theorem holds for anti-ferromagnetic Heisenberg chains, which rigorously justifies the numerical findings in . Moreover, as is exponentially large as increases, it can be used as a resource for universal quantum computation. For instance, we can encode qubits as , thereby performing universal quantum computation in . This is a remarkable observation: we have found a system where quantum computation can be achieved with a single switch, and where both the system and control Hamiltonian are physical, e.g. consist of nearest-neighbor two-body interactions, which are very common in physics. It provides possibly the simplest and most elegant way of achieving quantum computation so far (leaving efficiency issues beside ). Having only a single switch we avoid the experimental difficulty of quickly changing field directions.
Iii-B XYZ chain
For XYZ chain under control with
where , does the subspace controllability still exist? As discussed in Example 1, there exists a parity symmetry satisfying , with two invariant subspaces and , corresponding to eigenvalues of . We will show that the Hamiltonians cannot be further block-diagonalized on each of the two subspaces, and the system is controllable on each of them. Notice that, compared to the XXZ chain, the number of invariant subspaces for XYZ chain has reduced from to , which is not too surprising as we have broken the symmetry between X and Y directions from XXZ to XYZ type, and some symmetries should disappear. In the following we will identify all operators in generated by and :
Continuing this process, we obtain all , , and . Next, we have
and together with we can decouple and get
and . Similarly we can decouple and
independently generate and . This a major
difference from the XXZ case, where the XX and YY operators at
neighboring locations cannot be decoupled. Due to such decoupling, we
expect that the dynamical Lie algebra generated by and
will be larger than the XXZ case. Next, repeating the same generation
process by calculating the commutators, we get the following set series
of -body operators:
(1) : ;
(2) : , where can be or , and ;
(3) : and
() : when is even, we can generate:
When is odd, we can generate:
Compared with XXZ chain, where we can only generate the coupled Z-type operator, such as , for XYZ chain, we can separately generate and . is be divided into two subsets: the set of -type operators and the set of -type operators, where each operator can contain number of ’s and number of ’s, . Hence, following some basic combinatorics argument, we have:
and the dimension of ():
where we have used the identity
Since and are simultaneously block-diagonalized on , must be a subalgebra of . Moreover, since the -body operators in are generated from the -body operators, does not include two -type operators, the identity and , which are however included in . Hence, we have
Hence achieves the allowed maximal value, which is only true when restricted on each of the subspace and , or . Noticing that and are trace-zero on and for , we must have on both and for . When , it is easy to check that on and . Thus, we have proved the following theorem:
For an XYZ chain of length with a single local control on the end spin in direction, the system is controllable on each of the two invariant subspaces and .
Iii-C When Control Has a Leakage on Neighboring Spins
The previous assumption of control on a single spin only holds in theory. In practice, it is difficult to apply a control field that only acts on a single spin without affecting its neighbors. Hence, a more realistic assumption is that the local end control has a leakage on the neighboring spins with . We consider two common types of leakage: linear and exponential decays. In the following, we show that the subspace controllability results discussed so far are robust against such control leakage, i.e., when single control field has some leakage on the neighboring spins, the system is still controllable in the invariant subspaces. Under the leakage assumption,
Defining adjoint action of on as and , we have
where the coefficients in this expression can be denoted by
(1) When the leakage of the local control is linear, i.e., for different and , we can generate the operator from any two rows of . Analogously, we can generate and hence generate . From , we can also generate . From and , we can sequentially generate and , .
(2) When the leakage of the local control decays nonlinearly, e.g., , we have , and from the property of Vandermonde matrix, . Hence we can generate each , . Together with , we can decouple and generate , .
Hence, in both cases, generated by and in (6) is the same as that generated by and . In general, for other types of nonlinear leakage, the above reasoning is valid for almost all choices of . Thus we have:
For an XXZ or XYZ chain of length , under a single local control on the end spin in Z-direction with leakage to the neighboring spins, the system is controllable on each of the invariant subspaces.
Iv Minimal Controls for Full Controllability
In previous section, we have provided a complete discussion of the control problem of spin chains with the least control degree of freedom, i.e., a single local control at the end of the spin chain. In general, as the number of controls increases, existing symmetries will disappear and the system will become fully controllable on the entire Hilbert space under a sufficient number of independent controls. Therefore, another interesting theoretical question is to ask when such transition happens from an uncontrollable system to a fully controllable one. Alternatively, we can ask what are the minimal controls that can make the chain fully controllable, which is the main topic of this section. We will base on the results in previous discussions and add more controls to the control systems under (3) or (5).
Iv-a Controlling and
In , it was proved by the propagation property that an XXZ chain with two independent controls and is fully controllable on the entire space. We can rederive this result from our analysis in previous section: observing the operators generated by and , and writing down the operators generated by and , it is easy to see that we can generate all -body Pauli operators, in . Hence the system is fully controllable.
For an XXZ or XYZ chain of length , with two local controls on the end spin, and , the system is controllable on the whole space.
Iv-B Controlling and
In Theorem 4, we have shown that if we can fully control the end spin, then the system is controllable on the whole space. What if we can fully control one spin at other locations? We will prove that for a general XYZ chain two independent controls on the th spin in Z and X directions
are sufficient for controllability on the whole space, except when , where the Hamiltonians exhibit a mirror permutation symmetry with respect to the center th spin. Specifically, for , let us calculate the operators in generated by the three Hamiltonians.