Los Alamos National Lab 2019 Quantum Computing Summer School Introduction to Hamiltonian Complexity Elizabeth Crosson University of New Mexico Many-Body Quantum Physics The energy of a quantum mechanical system is an observable that corresponds to a Hermitian operator H called the Hamiltonian. H generates the time evolution of the system by the Schrodinger equation. By a “many-body quantum system” we mean a quantum system with interacting particles (e.g. spins, fermions, bosons, etc) in the regime where is large. The axioms of QM tell us that the full system Hilbert space is a tensor product of single particle Hilbert spaces: The limit is called the thermodynamic limit. Often we describe the asymptotic scaling of physical quantities with by “big-O” notation. is if for all sufficiently large n, g(n) is no larger than a constant times f(n). Ground States and Local Hamiltonians Many-body quantum systems can be extremely complicated, but if they are thermal equilibrium at low temperature then their properties are determined by ground states (the states of lowest energy): Where is the degeneracy of the ground space. Often without symmetry and the GS is unique. Another simplifying observation is that interactions in nature are few-body i.e. involve a few particles at a time. For example, in a classical system of many charged particles the interactions are governed by Coulomb's law which describes a potential between two charges at a time, and we sum these interactions to get the full energy. We restrict our attention to local Hamiltonians with . Precisely speaking, the local terms have the form , where acts on at most k particles. Leaving these identity factors implicit simplifies the notation. Locality: Spatial vs Combinatorial The word “locality” is overloaded. We just called Coulomb’s law “local”, but of course it is also acts far across space, decaying in strength with the square of the distance. To be more precise, a system is spatially local if the interactions involving each particle are confined to Euclidean ball of radius around the particle. Spatially local systems are important in the quantum description of matter (e.g. crystals, magnets, conducting metals), which often involves spins / bosons / fermions interacting with near-neighbors on a lattice: But we can also consider more general kinds of connectivity, where connections are long-range but still involve a few particles at a time. To be precise this could be called “combinatorial locality”. One justification for considering combinatorially local Hamiltonians that are not spatially local is that we can simulate the time evolution of such systems using a quantum computer. Examples of Local Hamiltonians The [classical, 1D, ferromagnetic] Ising model describes a chain of spin ½ particles on a line whose interaction energetically favors alignment along the Z direction: The ground space of this model can be determined by inspection to be . This is essentially a classical model so all the local terms commute and can be simultaneously diagonalized in a tensor product basis (the computational basis) and the ground space is spanned by unentangled states. We can make this model more interesting by adding a transverse field in the X direction. This is called the 1D ferromagnetic transverse Ising model: This 1D ferromagnetic TIM is more distinctly quantum. The local terms no longer commute, the ground state is quite entangled, and the analytical solution was an important milestone result in mathematical physics. Examples of Local Hamiltonians 1D spin models are often called “spin chains.” An example of a distinctly quantum spin chain whose ground state is relatively easy to analyze is the ferromagnetic Heisenberg model: The ground states of this model maximize the sum of the total angular momentum, And so (with a bit of work) one sees that the ground space is the symmetric subspace, spanned by the states: Where . To read this note that is the set of n-bit strings, and here we use to denote the Hamming weight of the string x (the number of 1’s in the string). Note that the ground states with m = 0 and m = n are product states, while for m = n/2 the ground state is entangled (i.e. spins in disjoint regions of the chain are entangled). Hamiltonian Complexity Analytical solutions describing ground states of strongly interacting many-body quantum systems are extremely rare. This motivates the use of classical and quantum computers in finding ground states. The field of Hamiltonian complexity characterizes the difficulty of finding ground states of local Hamiltonians using the theoretical tools of computational complexity. This is done using a formal model of computation that allows us to describe the asymptotic time and space requirements used to solve these problems. Using a formal notion of computational reduction (which tells us when a problem A is at least as difficult as another problem B) we will define complexity classes, such as P and NP, and we will relate the hardness of local Hamiltonian problems to other problems that appear in contexts which are distinct from quantum physics. Constraint Satisfaction Problems In part to establish some notation and terminology, we begin with a classical subset of local Hamiltonian problems called Boolean constraint satisfaction problem. Consider a collection of Boolean variables, In the Boolean context, we think of “1” as true and “0” as false. We express propositions (statements) by combining these variables with Boolean functions like AND (“ “), OR (“ “), and NOT (“ “). For example, is read “ and “ , which is true if is true and is true, and it is false otherwise. Any Boolean function can be described by a truth table (a list of inputs and outputs). By combining these basic functions we can form more complicated propositions e.g. Constraint Satisfaction Problems A generalized SAT problem asks us to decide whether there exists (“ “) an assignment to the variables which suffices to make a collection of propositions true simultaneously: Here is the i-th proposition, which is a statement about the k variables . Generalized SAT is the problem of deciding the truth value of existentially quantified Boolean formulas of the above form. If each is an OR statement involving variables or their negations, e.g. Then the existential formula is said to be in “conjunctive normal form” and the problem is called k-SAT, which may be more recognizable than generalized SAT. What is an upper bound on the time needed to solve generalized SAT ? Boolean CSPs as Local Hamiltonians We can also express Boolean formulas using our notation for quantum systems by promoting the Boolean variables to qubits . In this setting we construct a Hamiltonian with one local term for each in the Boolean formula. We want to map satisfying assignments to low energy states, so will have energy 0 when is satisfied. If we have a 3-SAT problem then each clause forbids one configuration e.g. Therefore our Hamiltonian assigns a higher energy to spin configurations that include this assignment: So that is a many-body Hamiltonian with energy corresponding to the number of violated clauses in the assignment . Therefore the ground energy of H is zero if and only if the formula is satisfiable. Boolean Circuits The elementary Boolean connectives AND, OR, and NOT also form a universal set of logic gates for Boolean circuits e.g. AND In this example, the inputs are , the output is c, and a and b are intermediate gate outputs. Universality means that arbitrary Boolean functions can be expressed using such Boolean circuits. If the circuit outputs a single bit then we call it a Boolean verifier circuit. The formal model of classical computation we consider today is based on Boolean verifier circuits (instead of Turing machines) because they generalize more naturally to the quantum setting. Problems and Languages We are building up towards formal definitions of computational complexity classes. For example, P is informally the set of problems that can be solved in a time that scales polynomially with the size of the problem description. This is regarded as the set of problems that can be solved efficiently by a deterministic classical computer. Before we can formally define P, we need to formally define our notion of what a computational problem is. This is done using the notion of a (formal) language. Definition (language): consider an alphabet (e.g. ) and the set of all strings of any length formed by letters of , e.g. A subset L of is called a language. Languages are very general, for example they could describe the set of all 3-SAT instances with a satisfying assignment, or the set of all local Hamiltonians with ground state energy 0, or the set of triangles with area A, etc. Circuit Verifiers Definition (Circuit Verifier): a circuit verifier is a family of Boolean circuits satisfying: 1. Each circuit has size r (size = number of gates in the circuit) 2. The output of each is a single bit, representing YES/NO or accept/reject. 3. (uniformity) There is a simple rule to go from the pattern of gates in to those in As is common in many references, we state property (3) informally to save time. The point of property (3) is to prevent us from embedding the hardness of the problem in the computation of . We don’t notate the input size of the verifiers at this stage, because the relation of r to the input size will be crucial to defining different complexity classes. Classical Complexity Theory Definition (polynomial time). if there exists a polynomial p and a verifier such that Here L is a language and |x| is the length of a string. P is regarded as the set of problems (languages) that can be efficiently solved (recognized) with a deterministic classical computer.