The most popular model of quantum computation is the quantum circuit model using single and two qubit gates as elementary transformations. Definitions of quantum complexity usually refer to this model. In contrast, we consider the physical interactions among the qubits as the basic computational resource that allows to carry out complex transformations on the quantum register. One method to compare the computational power of different interactions is to examine the complexity of the so-called mutual simulation of interaction Hamiltonians. The physical interaction can simulate other interactions by interspersing the natural time evolution with external control operations. In previous papers we have considered mutual simulation of n-partite pair-interaction Hamiltonians. We have focussed on the running time overhead of general simulations, while considering the required number of time steps only for special cases (decoupling and time-reversal). These two complexity measures differ significantly. Here we derive lower bounds on the number of time steps for simulations of general interaction graphs. In particular, the simulation of interaction graphs with irrational spectrum requires at least n steps. We discuss as examples graphs that correspond to graph codes and nearest neighbor interactions in 1- and 2-dimensional lattices. In the latter case the lower bounds are almost tight.
The most popular model of quantum computation is the quantum circuit model using single and two qubit gates as elementary transformations. Definitions of quantum complexity usually refer to this model. In contrast, we consider the physical interactions among the qubits as the basic computational resource that allows to carry out complex transformations on the quantum register. One method to compare the computational power of different interactions is to examine the complexity of the so-called mutual simulation of interaction Hamiltonians. The physical interaction can simulate other interactions by interspersing the natural time evolution with external control operations. In previous papers we have considered mutual simulation of n-partite pair-interaction Hamiltonians. We have focussed on the running time overhead of general simulations, while considering the required number of time steps only for special cases (decoupling and time-reversal). These two complexity measures differ significantly. Here we derive lower bounds on the number of time steps for simulations of general interaction graphs. In particular, the simulation of interaction graphs with irrational spectrum requires at least n steps. We discuss as examples graphs that correspond to graph codes and nearest neighbor interactions in 1- and 2-dimensional lattices. In the latter case the lower bounds are almost tight.
In previous papers we have considered mutual simulation of n-partite pair-interaction Hamiltonians. We have focussed on the running time overhead of general simulations, while considering the required number of time steps only for special cases (decoupling and time-reversal). These two complexity measures differ significantly. Here we derive lower bounds on the number of time steps for general simulations. In particular, the simulation of interaction graphs with irrational spectrum requires at least n steps. We discuss as examples graphs that correspond to graph codes and nearest neighbor interactions in 1- and 2-dimensional lattices. In the latter case the lower bounds are almost tight.
We consider a quantum computer consisting of n spins with an arbitrary but fixed pair-interaction Hamiltonian and describe how to simulate other pair-interactions by interspersing the natural time evolution with fast local transformations. Calculating the minimal time overhead of such a simulation leads to a convex optimization problem. Lower and upper bounds on the minimal time overhead are derived in terms of chromatic indices of interaction graphs and spectral majorization criteria. These results classify Hamiltonians with respect to their computational power. For a specific Hamiltonian, namely \sigma_z\otimes\sigma_z-interactions between all spins, the optimization is mathematically equivalent to a separability problem of n-qubit density matrices. We compare the complexity defined by such a quantum computer with the usual gate complexity.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner.However, such an implementation of Trotter steps has gate complexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques.We develop methods to perform faster Trotter steps with complexity sublinear in the number of terms.We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law.Our methods include one based on a recursive block encoding and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques.We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank.Combining with a tighter error analysis, we show that it suffices to use η 1/3 n 1/3 + n 2/3 /η 2/3 n 1+o(1) gates to simulate uniform electron gas with n spin orbitals and η electrons in second quantization in real space, asymptotically improving over the best previous work.We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation.We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic n-qubit two-local Hamiltonians with commuting terms require at least (n 2 ) gates to evolve with accuracy = (1/poly(n)) for time t = ( ).Our proof is based on a gateefficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-2 subspace, which may be of independent interest.Our result thus suggests the use of Hamiltonian structural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.
What is the time-optimal way of using a set of control Hamiltonians to obtain a desired interaction? Vidal, Hammerer, and Cirac [Phys. Rev. Lett. 88, 237902 (2002)] have obtained a set of powerful results characterizing the time-optimal simulation of a two-qubit quantum gate using a fixed interaction Hamiltonian and fast local control over the individual qubits. How practically useful are these results? We prove that there are two-qubit Hamiltonians such that time-optimal simulation requires infinitely many steps of evolution, each infinitesimally small, and thus is physically impractical. A procedure is given to determine which two-qubit Hamiltonians have this property, and we show that almost all Hamiltonians do. Finally, we determine some bounds on the penalty that must be paid in the simulation time if the number of steps is fixed at a finite number, and show that the cost in simulation time is not too great.
We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.