Abstract
We prove that the theorems of TDDFT can be extended to a class of qubit Hamiltonians that are universal for quantum computation. The theorems of TDDFT applied to universal Hamiltonians imply that singlequbit expectation values can be used as the basic variables in quantum computation and information theory, rather than wavefunctions. From a practical standpoint this opens the possibility of approximating observables of interest in quantum computations directly in terms of singlequbit quantities (i.e. as density functionals). Additionally, we also demonstrate that TDDFT provides an exact prescription for simulating universal Hamiltonians with other universal Hamiltonians that have different and possibly easiertorealize twoqubit interactions. This establishes the foundations of TDDFT for quantum computation and opens the possibility of developing density functionals for use in quantum algorithms.
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Introduction
The pioneering work of Hohenberg and Kohn^{1} in 1964 showed that the properties of a manybody system can be obtained as functionals of the simple electron density rather than the manybody wavefunction. Twenty years later, similar theorems were proven for timedependent systems^{3}. These developments have enabled complex simulations of physical systems at low computational cost using a very simple quantity. Can these ideas be extended to the domain of quantum computation and therefore enable similar progress in that field? In the present work, we prove analogous theorems to those of timedependent density functional theory (TDDFT) for the domain of universal quantum computation. In a similar spirit to TDDFT for electronic Hamiltonians, the theorems of TDDFT applied to universal Hamiltonians allow us to think of singlequbit expectation values as the basic variables in quantum computation and information theory, rather than the wavefunction. From a practical standpoint this opens the possibility of approximating observables of interest in quantum computions directly in terms of singlequbit quantities (i.e. as density functionals). Additionally, we demonstrate that TDDFT provides an exact prescription for simulating universal Hamiltonians with other universal Hamiltonians which have different and possibly easiertorealize twoqubit interactions. The theorems of TDDFT for universal Hamiltonians establish that TDDFT can in principle be used to simplify quantum computations, similar to how it has been applied in revolutionizing the simulation of atomic, molecular and condensed matter electronic structure dynamics. As we discuss below, the development of accurate approximate functionals for quantum simulation will be a necessary second step for the practical application of TDDFT to quantum computation.
We begin by briefly reviewing TDDFT for a system of Nelectrons described by the Hamiltonian
where and are respectively the position and momentum operators of the ith electron, is the electronelectron repulsion and ν(r, t) is a timedependent onebody scalar potential which includes the potential due to nuclear charges as well as any external fields. is the electron density operator, whose expectation value yields the oneelectron probability density. The first basic theorem of TDDFT, known as the “RungeGross (RG) theorem”^{3}, establishes a onetoone mapping between the expectation value of and the scalar potential ν(r, t) and therefore through the timedependent Schrödinger equation, a onetoone mapping between the density and the wavefunction. The RG theorem implies the remarkable fact that in principle, the oneelectron density contains the same information as the manyelectron wavefunction. The second basic TDDFT theorem known as the “van Leeuwen (VL) theorem”^{4} gives a prescription for constructing an auxiliary system with a different and possibly simpler electronelectron repulsion , which simulates the density evolution of the original Hamiltonian in Eq. 1. When , this auxiliary system is referred to as the “KohnSham system”^{2} and due to it's simplicity and accuracy, is in practice used in most DFT and TDDFT calculations.
It is not obvious that the RG and VL theorems extend to qubits, which are distinguishable spin 1/2 particles. In the results section, we prove analogous RG and VL theorems for a system of N qubits described by the very general universal 2local Hamiltonian^{5,6},
Here, are Pauli operators for the ith qubit, h_{i}(t) are local applied fields arbitrarily chosen along the zaxis and and are twoqubit interaction terms respectively parallel and perpendicular to the direction of the fields. The above Hamiltonian describes an open chain of N qubits arranged in a onedimensional array, with each qubit interacting with its nearest neighbors.
More general geometries are discussed in the supplementary material. In Refs.^{5,6}, it was shown that by appropriately tuning the local fields in Eq. 2, one can use the fixed twoqubit interaction alone to realize a set of universal twoqubit and singlequbit quantum gates, which in turn can be employed to perform universal quantum computation. In Eq. 2, the case where yields the Heisenberg Hamiltonian which describes exchange coupled spins in solid state arrays or quantum dots in heterostructures^{7}. The situation and yields the XXZ Hamiltonian, used to model electronic qubits on liquid Helium^{8} or solidstate systems with anisotropy due to spinorbit coupling^{9}, while the limit yields the XY model describing superconducting Josephson junction qubits^{10}. In the forthcoming sections, we will develop the TDDFT theorems for the Hamiltonian in Eq. 2 and discuss their implications for quantum computation and information theory.
Results
The qubit RungeGross theorem for quantum computation
We now state the equivalent RG theorem for quantum computation with the Hamiltonian in Eq. 2, the qubit RungeGross (qRG) theorem:
Theorem  For a given initial state ψ(0)〉 evolving to ψ(t)〉 under the Hamiltonian in Eq. 2 and with and fixed, there exists a onetoone mapping between the set of expectation values and the set of local fields {h_{1}, h_{2},…h_{N}} up to a constant global field (see supplementary information), over a given interval [0, t].
Here, we have defined as the expectation value of the component of the ith qubit along the field direction (zaxis). A detailed proof together with a more rigorous discussion of the conditions on the theorem are provided in the supplementary material. The qRG theorem implies that the set of local fields can be written as unique functionals of the set of expectation values , as illustrated in the first part of Figure 1. Since the solution to the timedependent Schrödinger equation is unique and and and are fixed, the wavefunction is a unique functional of the local fields. i.e. ψ(t)〉 ≡ ψ[h_{1}, h_{2}, …h_{N}](t)〉, where the square brackets denote that ψ is a functional of the set {h_{1}, h_{2}, …h_{N}} over the interval [0,t]. This fact, combined with the qRG theorem allows us to state a corollary, which is the first central result of this paper:
Corollary  There exists a onetoone mapping between the set of expectation values over the entire interval [0,t] and the Nqubit state ψ(t)〉.
The above corollary implies the counterintuitive fact that the full Nqubit wavefunction, which lives in a 2^{N} dimensional Hilbert space, is a unique functional of only the N components of each qubit along the zaxis over the interval [0,t]. i.e.
This naturally implies that no two wavefunctions evolving under the Hamiltonian in Eq. 2 can give the same set of expectation values for the entire timeinterval [0,t]. Having established the qRG theorem, we now proceed to discuss its implications for quantum computation.
Implications of the qubit RungeGross theorem for quantum computation
Although the qRG theorem does not tell us an explicit functional form for ψ, it has profound conceptual implications from a quantum information perspective. At first glance, it might appear that the set contains much less information than the full wavefunction, since projective measurements needed to obtain would seem to imply that information about noncommuting observables, or observables depending on multiqubit correlations is lost. However, since the wavefunction completely specifies all properties of the system, Eq. 3 implies that even properties depending on noncommuting observables or multiqubit correlations, such as entanglement and phase information are in fact uniquely determined by the set of expectation values .
From a practical standpoint, the qRG theorem implies that all observables can directly be constructed as functionals of singlequbit expectation values, without regard for the wavefunction. Although the qRG theorem proves that the set of expectation values in principle contains all of the quantum information in ψ, extracting this information in the form of a functional of is not always straightforward. In order to do this, one must either guess the exact functional form of the observable, or try to approximate it. Borrowing an analogy from electronic TDDFT, the timedependent dipole moment is a very simple density functional, while the average momentum of the system is not simple to construct as an explicit density functional, since it depends on the density very nonlocally in both space and time^{11}. A density functional for the average momentum must therefore be approximated in practical applications.
In quantum computation and information theory, a similar situation arises. Often, the observable of interest is simply a subset of on designated readout qubits which encode the answer to the computation and this subset is trivially a functional of the entire set. For instance, a simple example is the DeutschJozsa algorithm, where one measures a subset of in a query register to determine if a function f(x) is constant or balanced^{12}. If one finds the spin density of this subset to be zero everywhere, f(x) is constant, while if it is nonzero, f(x) is balanced. A more challenging observable functional to construct is twoqubit entanglement. We find that an exact pure state entanglement functional can in fact be constructed for a computation in which the state space is restricted to states where . The pure state entanglement (as measured by concurrence^{13}) between any two qubits labeled k and l can be written as a functional of the set for this particular case as (the derivation is provided in the supplementary material)
Interestingly, this particular entanglement functional is timelocal, since it depends only on the set at a given instant in time and so . In the more general case, observables may be nonlocal in time and depend on the set over an entire interval [0,t]. Although the functional in Eq. 4 is timelocal, it is “spatially” nonlocal, since the entanglement between qubits k and l depends on the components of all of the other N – 2 qubits. If one considers two flipped qubits instead of one, the entanglement functional becomes complicated and nonlocal in both space and time due to dependence on phases in the wavefunction (see supplemental material). Understanding the spatial and temporal nonlocality of density functionals in electronic structure theory is a very active research topic^{14,15} and interestingly a similar situation arises here in TDDFT for quantum computation as well.
Thus far we have proven the qRG theorem, which establishes that all observables of an Nqubit system can be obtained directly from the set of singlequbit expectation values , without needing explicit access to the wavefunction. However, in order to make this fact useful from a practical standpoint, one would like to be able to obtain the set by solving an auxiliary problem that is simpler than obtaining ψ(t)〉 itself. In the next section, we prove that there are in fact infinitely many universal Hamiltonians which can be used to simulate the same set and by choosing a Hamiltonian with a simpler evolution, one can in fact make TDDFT a practical tool for quantum computation.
A theorem analogous to the Van Leeuwen theorem for quantum computation
We now turn to the second fundamental theorem of TDDFT for universal computation, a VLlike theorem for qubits, the qubit Van Leeuwen theorem (qVL):
Theorem  Consider a given set of spin components obtained from the wavefunction ψ(t)〉 evolved under the Hamiltonian in Eq. 2. One can always construct (see supplementary material for certain conditions) a Hamiltonian with different twoqubit interactions denoted and and different local fields , which evolves a possibly different initial state ψ′(0)〉 to a different final state ψ′(t)〉 such that the condition is satisfied on the interval [0,t].
Here, we have defined . The qVL theorem allows us to obtain the set by simulating the evolution with an auxiliary Hamiltonian having different twoqubit interactions and hence a different (and possibly simpler) wavefunction evolution as illustrated in Figure 2. Furthermore, the qVL theorem guarantees that the auxiliary fields , are unique functionals of the set . As we discuss in the next section, this fact opens the possibility of simplifying computations by constructing simple approximations to the auxiliary fields as functionals of singlequbit expectation values. This is a similar concept to how the exchangecorrelation potential of electronic TDDFT is approximated as a functional of the onebody density in the KohnSham scheme.
A numerical demonstration of the qubit Van Leeuwen theorem
Before discussing general approximate functionals for the auxiliary local fields , in this section we will demonstrate the qVL theorem by constructing the exact functional for a simple example where an exact numerical solution is possible. The proof of the qVL theorem gives a mathematical procedure (see supplementary material) for engineering the exact auxiliary fields which reproduce a given set under a different twoqubit interaction. As a simple demonstration, we use this procedure to numerically simulate a 3qubit Heisenberg Hamiltonian using an XY Hamiltonian as the auxiliary system (Figure 3). For the simulation, the system is prepared in the initial state , where 1〉 and 0〉 are eigenstates of with eigenvalues −1 and 1 respectively. In the Heisenberg Hamiltonian, and we choose J_{12} = J_{23} = 0.5, which represents a chain with isotropic and uniform antiferromagnetic couplings. We apply a pulse of the form (odd harmonics) to the first qubit and (even harmonics) to the third qubit. The timedependent Schrödinger equation is solved numerically and the set is read out during the evolution. Details of the simulation are provided in the supplementary material.
For the auxiliary XY Hamiltonian, and we choose different and nonuniform couplings in which and . Thus, we have chosen the auxiliary system to be anisotropic, with nonuniform and alternating ferromagnetic and antiferromagnetic couplings. Using the qVL theorem, we engineer the auxiliary local fields which using a this XY interaction, reproduce the set obtained from the original evolution under the uniform Hesienberg Hamiltonian. As seen in Figure 3, the auxiliary local fields are quite different from the original local fields applied to the Heisenberg model, but simulate the set of components correctly. i.e. = . In the language of electronic TDDFT, the XY model in our simulation is analogous to the “KohnSham system” and the set play the role of the exact KohnSham potential as a density functional.
In the above example, we have constructed the exact auxiliary fields a posteriori, after having already solved the wavefunction evolution of the original system. Although such exact solutions are valuable in guiding functional development, one would ultimately like to develop accurate approximate and generic functionals for the auxiliary fields which can be used to circumvent solving the original problem. Furthermore, one would like to choose the auxiliary system so that its evolution is simpler than that of the original system. Such an approach has proven invaluable in the KohnSham scheme of electronic TDDFT and we now discuss its applicability to TDDFT for quantum computation.
Discussion
The qRG and qVL theorems place TDDFT for universal quantum computation on a firm theoretical footing and open several exciting research avenues. The development of approximate density functionals has been essential for the success of electronic TDDFT and will be in quantum computation and information theory as well. In the KohnSham scheme of electronic TDDFT, one simulates the correlated manybody system evolving under the Hamiltonian of Eq. 1, with an uncorrelated noninteracting system in which w′(r_{i} – r_{j}) = 0. The effective “KohnSham” potential v′(r,t) of this noninteracting system must be approximated as a functional of the density. The local density approximation (LDA)^{2}, was the first density functional to be applied to solidstate systems in the 1960s, but it was not sufficiently accurate for quantum chemistry. More than 20 years elapsed between the fundamental DFT theorem of Hohenberg and Kohn^{1} and the development of density functionals capable of achieving chemical accuracy in the 1980's; the so called generalized gradient approximations (GGA's)^{16}.
In a similar vein, although we have established the fundamental theorems of TDDFT for quantum computation, the development of accurate approximate functionals will be a future challenge. Additionally, in TDDFT for quantum computation, we expect the path of functional development to be somewhat different. In the electronic Hamiltonian (Eq. 1), the kinetic and electronelectron repulsion are always the same operators and similarly the KohnSham system is always noninteracting. Therefore, the KohnSham potential is always the same functional for any electronic system. In contrast, in quantum computation one uses different twoqubit interaction terms depending on which universal Hamiltonian implements a given quantum circuit and therefore the functional will be different for each situation. For instance, if one wants to simulate an antiferromagnetic Heisenberg model using a ferromagnetic Heisenberg model, the functional will be different than a simulation of the same system using an XY model. Therefore, functional development will need to focus on specific implementations of quantum algorithms, rather than a single universal functional for all quantum computations. Typically, one would want to choose the auxiliary system's wavefunction to be less entangled than that of the original system, thereby making it easier to simulate using TDDFT on a classical computer. This is a similar concept to how TDDFT has been applied to electronic systems, where TDDFT provides a tool to approximately simulate quantum manybody systems efficiently on classical computers.
Naturally, there are systems that will be very hard to simulate using approximate functionals, such as those that are in the complexity class QMA and may require exponentially scaling resources on a quantum computer^{30}. The collapse of the computational complexity class hierarchy is of course not expected and therefore finding functionals that carry out complex quantum computational tasks is extremely unlikely. Nevertheless, understanding how TDDFT functionals can approximately simulate efficient quantum algorithms on a classical computer is an open direction. Density functionals for strongly correlated lattice and spin systems have been recently proposed^{17,18,19,20} and could be applied to several problems of relevance in quantum computing. In Refs. ^{17,18,19,20} local density (LDA) and generalized gradient approximations (GGA) for one dimensional Hubbard chains and spin chains were derived from exact Bethe ansatz solutions and could readily be applied to solidstate quantum computing or perfect state transfer protocols in spin networks^{21}. Functionals can also be parametrized from numerical simulations of onedimensional qubit systems using timedependent density matrix renormalization group methods (TDMRG)^{22}, in an analogous fashion as quantum Monte Carlo simulations of the uniform electron gas have proven invaluable in electronic DFT^{23}. In Figure 4, we summarize the analogies between electronic TDDFT and TDDFT for quantum computation, which will necessarily giude development of approximate functionals.
It should be noted that at present, the existing density functionals used in electronic structure calculations are far too simple to capture the entanglement and subtle correlations that play a major role in most quantum computing schemes. For instance, the adiabatic LDA and GGA functionals mentioned above are local in time and local or semilocal in space. As a result, they are poorly suited to systems that are strongly correlated and highly entangled as is typically the case in quantum computations. Whether or not it is possible to develop sufficiently nonlocal functionals for quantum computations remains an open question and is an essential prerequisite for making the theorems we have proven practically useful.
In a different direction, one could also imagine using the qVL theorem as an experimental tool to engineer different physical systems which perform the same computations. For instance, one could simulate an algorithm on an ion trap using a system of superconducting flux qubits, by using the qVL theorem to engineer the flux qubit Hamiltonian from knowledge of how the algorithm is performed on the ion trap. Another important research direction will be the generalization of DFT and TDDFT to other universal Hamiltonians and models of quantum computation. For instance, Ref. ^{24} discussed the use of TDDFT for obtaining gaps in adiabatic quantum computation. In^{29}, groundstate DFT was used to study relationships between entanglement and quantum phase transitions, while Ref. ^{30} explored DFT from a complexity theory perspective.
In the supplementary material we explore connections between TDDFT for quantum computation and lattice theories of TDDFT^{25,26,27,28}.
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Acknowledgements
Useful discussions with S. Mostame, J. D. Whitfield, S. Boxio, M. H. Yung and J. Parkhill are greatfully acknowledged. We thank NSF award PHY0835713 for financial support.
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D. G. T. and A. A. G. both developed the theory, performed the calculations and also wrote the manuscript.
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Tempel, D., AspuruGuzik, A. Quantum Computing Without Wavefunctions: TimeDependent Density Functional Theory for Universal Quantum Computation. Sci Rep 2, 391 (2012). https://doi.org/10.1038/srep00391
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DOI: https://doi.org/10.1038/srep00391
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