Time Evolution¶

Time Evolution in QPE¶

Scope¶

This document describes how time evolution is implemented for Quantum Phase Estimation (QPE) in this repository, including:

construction of the unitary $U = e^{-iHt}$
Trotter decomposition
controlled time evolution
numerical trade-offs and errors

Time evolution is the core computational primitive underlying QPE.

Overview¶

QPE requires implementing:

$$ U = e^{-iHt} $$

where:

$H$ is the molecular Hamiltonian
$t$ is the evolution time

This unitary encodes the energy eigenvalues as phases.

Hamiltonian Structure¶

After fermion-to-qubit mapping:

$$ H = \sum_j c_j P_j $$

where:

$c_j$ are real coefficients
$P_j$ are Pauli strings (e.g. $XZIY$)

Exact Evolution (Ideal)¶

If all terms commute:

$$ e^{-iHt} = \prod_j e^{-i c_j P_j t} $$

However, in general:

$$ [P_i, P_j] \ne 0 $$

so exact factorization is not possible.

Trotter Decomposition¶

To approximate evolution, this repository uses Trotterization.

First-order Trotter formula¶

$$ e^{-iHt} \approx \left( \prod_j e^{-i c_j P_j t / r} \right)^r $$

where:

$r$ = number of Trotter steps

Error Behaviour¶

Trotter error scales as:

$$ \mathcal{O}\left(\frac{t^2}{r}\right) $$

Implications:

increasing trotter_steps reduces error
error grows with total evolution time $t$

Implementation (This Repository)¶

Hamiltonian source¶

Time evolution is built from:

H, n_qubits, hf_state = build_hamiltonian(...)

All Pauli terms in (H) are used to construct evolution.

Pauli exponentials¶

Each term:

[ e^{-i c_j P_j t} ]

is implemented using:

basis rotations
controlled rotations
inverse basis rotations

Controlled evolution¶

QPE requires:

[ U^{2^k} = e^{-iH t 2^k} ]

This is implemented by:

scaling evolution time:
- ( t \rightarrow t \cdot 2^k )
applying Trotterized evolution

Circuit Structure¶

For each ancilla qubit:

ancilla ──●────────────
           │
system  ───U^(2^k)────

This controlled evolution is repeated for each ancilla.

Parameters¶

Evolution time¶

--t

Controls:

phase resolution
energy scaling

Trade-offs:

larger (t):
- higher precision
- risk of phase wrapping
smaller (t):
- safer range
- lower resolution

Trotter steps¶

--trotter-steps

Controls:

approximation accuracy
circuit depth

Ancilla count¶

--ancillas

Controls:

phase precision
number of controlled evolutions

Combined Trade-offs¶

Parameter	Effect
(t)	↑ precision, ↑ wrapping risk
trotter_steps	↓ error, ↑ circuit depth
ancillas	↑ resolution, ↑ circuit size

Noise Effects¶

Under noise:

each Trotter step introduces decoherence
deeper circuits amplify errors
controlled operations are especially sensitive

Practical implication¶

fewer Trotter steps → less noise, more approximation error
more Trotter steps → better approximation, worse noise

Practical Guidance¶

Baseline settings¶

t = 1–2
trotter_steps = 2–6

Improve accuracy¶

increase trotter_steps
increase ancillas
ensure good input state

Reduce noise impact¶

reduce circuit depth (lower trotter_steps)
reduce ancilla count
use smaller (t)

Limitations¶

first-order Trotter only
no higher-order Suzuki formulas
no adaptive Trotterization
no qubitization or advanced Hamiltonian simulation

Summary¶

Feature	Status
Trotter decomposition	First-order
Controlled evolution	Implemented
Shared Hamiltonian	Yes
Noise support	Yes

Key Takeaway¶

Time evolution in this repository is implemented using Trotterized exponentials of Pauli terms, providing a practical and flexible approximation of (e^{-iHt}) suitable for QPE workflows.

The design prioritizes:

simplicity
compatibility with the shared Hamiltonian pipeline
tunable accuracy via trotter_steps