Hamiltonian simulatioN

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Background

The Hamiltonian Simulation problem describes the evolution of quantum systems, such as molecules and solid state systems, by solving the Schrodinger equation. Quantum computers enable the simulation in a scalable manner, as described in [Lloyd96]. The most notable algorithm is the Trotterization-based product formula.

The Problem

Generate a circuit, using no more than 10 qubits, that approximates the unitary $e^{-iH}$ where $H$ is the qubit hamiltonian of a LiH (lithium hydride) molecule. The LiH Hamiltonian is composed of 276 Pauli strings, and can be found HERE. The approximation error is defined in the next section, and should be less than 0.1. The circuit should be composed of the CX and single qubit gates only.

Approximation Error

The “distance” between the approximated circuit and the real unitary operator is defined by the operator norm of the difference between the two operators

$error = \|U_{circuit} - e^{-iH}\|$.

The operator norm is defined as the maximal eigenvalue of the operator $\|A\| = max {|\lambda_i|}$ where $\lambda_i$ are the eigenvalues of $A$.

This error describes the worst inconsistency between the two operators for all possible input states.

Note that for unitary operators the error is bounded between 0 and 2.

Metric

The winning solution will be the circuit with the minimal depth for which the error defined above is less then 0.1.

Troterrization Algorithm

In this section we are going to describe the most common algorithm - Troterrization based product formula.
The algorithm relays on the Lie-Trotter formula

$$e^{-i(H_1+H_2+...+H_l)t}= \lim_{n\to\infty} (e^{-\frac{iH_1t}{n}}e^{-\frac{iH_2t}{n}}...e^{-\frac{iH_lt}{n}}) ^n. $$

This formula is especially useful in the context of quantum simulations since generally the Hamiltonian is given by a sum of Pauli terms, and a single Pauli term can be exponentiated as presented.

For a Pauli Z Hamiltonian, the unitary $e^{-iZZZt}$ can be presented using the following equivalent circuits:

An exponentiation of a general Pauli term (consist of I, X, Y, Z Pauli matrices), such as $e^{-iXZXt}$ is composed from $e^{-iZZZt}$ with basis change via single-qubit gates:

Next, we will observe a possibility to optimize the circuit for multi-term Hamiltonian. Note that for a multi-term Hamiltonian such as $H = H_1 + H_2$ the ordering of the terms within a single repetition of the trotter decomposition is irrelevant, i.e 

$$e^{-i(H_1 + H_2)t} =\lim_{n\to \infty} (e^{-\frac{iH_1t}{n}}e^{-\frac{iH_2t}n}) = \lim_{n\to \infty} (e^{-\frac{iH_2t}{n}}e^{-\frac{iH_1t}{n}})$$

even if $H_1$ and $H_2$ are not commutative.

These schemes allow compiling of shorter circuits. For example, consider the Hamiltonian $H = ZZZ + ZZ$. Naively the single repetition would compile to:

Using a different implementation for single term evolution:

some CX gates cancel out to get:

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