Hamiltonian Simulation (pt. 1)CSE 490Q: Quantum Computation

More Grover

• Quantum circuit takes

• The |-> part is left unchanged, so we can ignored it in the analysis• state G|ψ> is really G|ψ> ⊗|->

• Alternatively, you can think of the |-> bit as being hidden in the circuit• treat it as something like a “local variable” in Java code

Hamiltonian Simulation

• Simulating quantum physics

• Probably the most important application of quantum computers• e.g., simulating the behavior of molecules• potential uses include vaccine development and developing new

industrial processes that do not produce C02

• Laws describing physical behavior are often differential equations• describe the rate of change of the system• have to solve them to determine behavior

• Very simple case: x’(t) = a x(t)• rate of change is a constant times current value• solution is x(t) = eat x(0)

• (x(0) factor makes the sides equal when t = 0)

Physical Laws

Schrödinger’s Equation

• Schrödinger’s equation describes quantum mechanical systems

• Another simple case: |x’(t)> = -iH |x(t)>• rate of change is a constant times current value• but now the state is a vector rather than a scalar

• Answer should be |x(t)> = e-iHt |x(0)>

• Problem: what the heck is e-iHt ?• exponential of a matrix?

Functions with Matrix Arguments

• Let f(x) be a function with a real number input and output

• Suppose that f has a Taylor series expansion converging in (a, b)• That is, we have f(x) = a0 + a1 x + a2 x2 + a3 x3 + … for x in (a, b)

• Then we define f(X), with a matrix argument X, as

f(X) = a0 + a1 X + a2 X2 + a3 X3 + …

• We need to show that this converges to some matrix…

Functions with Matrix Arguments

• Suppose that X has an eigenvalue decomposition:

X = V diag(w1, …, wr) Vt

• We can see that

using the fact that diag(a1, ... ar) diag(b1, …, br) = diag(a1b1, …, arbr)

Functions with Matrix Arguments

• Suppose that X has an eigenvalue decomposition:

X = V diag(w1, …, wr) Vt

• More generally, we can show that Xk = V diag(w1k, …, wk

k) Vt by induction:

• We can calculate f(X) as follows:

using the fact that diag(a1, ... ar) + diag(b1, …, br) = diag(a1+b1, …, ar+br)

Functions with Matrix Arguments

• Suppose that f has a Taylor series expansion converging in (a, b):

f(x) = a0 + a1 x + a2 x2 + a3 x3 + … for x all in (a, b)

• Suppose that X has an eigenvalue decomposition:

X = V diag(w1, …, wr) Vt

• Then, we have seen that f(X) = V diag(f(w1), …, f(wr)) V• this converges provided that w1, …, wr all lie in (a, b)• so f(X) is defined for matrices with real eigenvalues in this interval

Functions with Matrix Arguments

Schrödinger’s Equation

• Time-dependent Schrödinger equation: |x’(t)> = -iH |x(t)>

• Answer is |x(t)> = e-iHt |x(0)>

• Makes sense if the Taylor series expansion of e-ix converges at H’s eigenvalues

• Good news! The Taylor series expansion of e-ix converges everywhere

Schrödinger’s Equation

• Time-dependent Schrödinger equation: |x’(t)> = -iH |x(t)>• Answer is |x(t)> = e-iHt |x(0)>

• If w is an eigenvalue of H, then e-iw is an eigenvalue of e-iH

• if w is a real, then |e-iw| = 1

• Hence, if all eigenvalues of H are real, then e-iH is unitary• (recall that unitary matrices are those with eigenvalues z s.t. |z| = 1)

• The solution makes sense only if H has real eigenvalues• H has real eigenvalues iff Ht = H, i.e., iff H is Hermitian

Hamiltonian Simulation

Problem: Given a Hermitian matrix H and initial state |x0> and a time t,prepare the state e-iHt |x0>

• Any reasonable Hamiltonian will be Hermitian

• Result is the solution of the time-dependent Schrödinger equation

• We can also just take t = 1• just rescale the Hamiltonian given by t

• (More general case is to allow the Hamiltonian to change with time.)

Hamiltonians and Unitaries

• If H is Hermitian, then e-iH is unitary

• Reverse is also true:• if U is unitary, then every eigenvalue is of the form z = e-iw

• this means that -i log(z) = -i log(e-iw) = -i (-iw) = -i2 w = w• hence, -i log U is defined (the Taylor series of log converges)

and its eigenvalues are real, so -i log U is Hermitian

Hamiltonians and Unitaries

• If H is Hermitian, then U = e-iH is unitary• every Hermitian is -i times the log of a unitary

• If U is unitary, then H = -i log U is Hermitian• every unitary is the exponential of -i times a unitary

• We are free to think either in terms of unitaries or Hamiltonians

• Every quantum circuit applies some unitary, so it simulates some Hamiltonian• in principle, every quantum computation is just Hamiltonian simulation!