Statistical Phase Estimation
In this tutorial, we will go over how to execute statistical phase estimation (SPE) using QURI Algo and QURI VM.
Overview
SPE is an algorithm that is used to estimate the eigenenergies of a problem Hamiltonian. It does so by performing a Hadamard test on a trial wave-function, resulting in a measurable signal that can be used to infer the wave-functions spectral density. If this wave-function has a significant overlap with the true ground state of the system, the ground state energy should make a significant contribution to the spectral density. The following propoerties makes SPE an attractive algorithm in EFTQC
- Good circuit depth scaling compared to FTQC algorithms
- The run-time scales with the energy accuracy as
- Noise resilience
In this notebook we will introduce all of the pieces needed to run SPE. In order, this notebook will
- Explain the LT22 and Gaussian filter variants of SPE and showcase their implementations in QURI Algo
- Instruct in the use of QURI Parts' noise models and showcase the robustness of SPE to noise based on the STAR architecture
- Estimate the Fidelity of the Hadamard test circuits used in SPE after transpilation to the STAR architecture and introduce QURI VM.
Prerequisites
We do not go into great detail about the following topics, but they will be part of this notebook.
Set up the system
Before running SPE we briefly introduce some of the building blocks we need. We will here show how to
- Define a molecule
- Set up programming abstractions needed for SPE using QURI Algo
- Verify that they behave as expected by conducting the Hadamard test
We first set up the problem using PySCF and quri-parts. PySCF is used to define the molecular geometry and then give us the spin-restricted Hartree-Fock ground state solution to the molecular Hamiltonian. Then the second quantized Hamiltonian is generated by calculating the electron integrals, which is then converted to a qubit operator. For this we use the quri_parts.openfermion
and quri_parts.pyscf
modules.
from pyscf import gto, scf
from quri_parts.pyscf.mol import get_spin_mo_integrals_from_mole
from quri_parts.openfermion.mol import get_qubit_mapped_hamiltonian
from quri_algo.problem import QubitHamiltonianInput
# Prepare Hamiltonian
mole = gto.M(atom="H 0 0 0; H 0 0 1")
mf = scf.RHF(mole).run(verbose=0)
hamiltonian, mapping = get_qubit_mapped_hamiltonian(
*get_spin_mo_integrals_from_mole(mole, mf.mo_coeff)
)
Before moving on, we compute some exact system characteristics, e.g. the exact ground state energy and the exact energy gap between the ground state and the first excited state. To this end, we first convert the Hamiltonian to a sparse matrix that we can diagonalize using scipy's sparse linear algebra library.
import numpy as np
from quri_parts.core.operator import get_sparse_matrix
vals, vecs = np.linalg.eigh(get_sparse_matrix(hamiltonian).toarray())
EXACT_GS_ENERGY = np.min(vals)
EXACT_GAP = np.abs(vals[np.argsort(vals)][:2] @ [1, -1])
print("E_0:", EXACT_GS_ENERGY)
print("Delta:", EXACT_GAP)
E_0: -1.1011503302326187
Delta: 0.3552785371970515
To perform further computations, we need to encode our problem into some circuit that represents some operator function of our problem. For example in SPE, to estimate the ground state of a Hamiltonian, we need to encode our Hamiltonian into a controlled time evolution circuit. Here, we choose to do the time evolution with Trotterization. To do this, we wrap the hamiltonian into QubitHamiltonianInput
for encoding into a circuit later.
QubitHamiltonianInput
can be seen as a generalization of the Operator
class from QURI Parts, containing both the terms of a qubit operator and the number of qubits. This makes it convenient when contructing a class hierarchy with quantum circuit factories intended to represent time-evolution and other problem defined quantum circuit constructs.
hamiltonian_input = QubitHamiltonianInput(mapping.n_qubits, hamiltonian)
Circuit factories
As a direct example of the aforementioned circuit factories we consider the TrotterControlledTimeEvolutionCircuitFactory
. As the name suggests this is a circuit factory that instance time-evolution circuits based on the Trotter-Suzuki decomposition. It needs a QubitHamiltonianInput
which serves as the generator of time evolution, as well as the number of Trotter steps. Details of this factory are explained in circuit factories.
from quri_parts.circuit.utils.circuit_drawer import draw_circuit
from quri_algo.circuit.time_evolution.trotter_time_evo import (
TrotterControlledTimeEvolutionCircuitFactory,
)
trotter_concotrolled_time_evo_circuit_factory = (
TrotterControlledTimeEvolutionCircuitFactory(hamiltonian_input, n_trotter=30)
)
# Time-evolution with t=1
c_time_evo = trotter_concotrolled_time_evo_circuit_factory(evolution_time=1)
draw_circuit(c_time_evo, line_length=10000)
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|PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |PR | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
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As seen above, the time-evolution is composed of a series of Pauli rotations. Since it is a controlled time-evolution, each term in the Hamiltonian results in two Pauli rotations, i.e.
where is the controlled Paui rotation gate with rotation generator .
We represent these Pauli rotations as single quantum gates, however to run them on a real device, in most cases, they need to first be transpiled to a series of two-qubit gates and single qubit rotations. Later in this notebook, we will show how this is done to conform to the STAR architecture.
Estimator
The SPE algorithm requires an esimtation of on different evolution time . Follow the estimator tutorial, we build the exact estimator and another one that based on Hadamard test with Trotterized time evolution operator.
from quri_algo.core.estimator.time_evolution_estimator.exact_spectrum import (
ExactTimeEvolutionExpectationValueEstimator
)
from quri_algo.core.estimator.time_evolution_estimator.trotter import (
TrotterTimeEvolutionHadamardTest
)
from quri_parts.qulacs.sampler import create_qulacs_vector_sampler
sampler = create_qulacs_vector_sampler()
exact_time_evolution_estimator = ExactTimeEvolutionExpectationValueEstimator(
hamiltonian_input, vals, vecs
)
trotter_time_evolution_estimator = TrotterTimeEvolutionHadamardTest(
hamiltonian_input, sampler, 50
)
We then build a state as an example. We choose a computational basis superposition state.
from quri_parts.core.state import quantum_state, comp_basis_superposition
state = comp_basis_superposition(
quantum_state(mapping.n_qubits, bits=0b0011),
quantum_state(mapping.n_qubits, bits=0b1100),
7 / 20,
0,
)
This state is chosen such that its overlap with the exact ground state vector is about 75%.
from quri_parts.qulacs.simulator import evaluate_state_to_vector
gs_overlap = np.abs(evaluate_state_to_vector(state).vector @ vecs[:, 0])**2
print(f"Overlap between the chosen state and the exact grounds state vector: {gs_overlap: .1e}")
Overlap between the chosen state and the exact grounds state vector: 7.5e-01
Let's compute with the state above. First we use the exact estimator, which is based on matrix multiplication
print(
f"The time-evolution expectation value is {exact_time_evolution_estimator(state, evolution_time=5.).value}"
)
print(
f"The trotter time-evolution expectation value is {trotter_time_evolution_estimator(state, evolution_time=5., n_shots=int(1e6)).value}"
)
The time-evolution expectation value is (0.7803970841187445-0.5734363633936669j)
The trotter time-evolution expectation value is (0.780258-0.574082j)
As we shall see below, this Hadamard test is essential to SPE. The tools provided above can be used to dictate whether the Hadamard test should be done exactly or with a Trotterized quantum circuit implementation based on sampling.
Executing statistical phase estimation
SPE has two main variants, which we will explore in this notebook. The first one is by Lin and Tong (LT22) which introduced SPE in the context of early fault tolerant devices for the first time. We also explore a variant of it by Wang et. al. which we refer to as Gaussian filter. The chief difference between the two is that the signal that is generated uses a convolution function that is either a Heaviside step functino (LT22) or a Gaussian (Gaussian filter).
Introducing the signal and convolution functions
SPE tries to obtain the spectrum of a Hamiltonian by using a trial state . `s overlap with the eigenstates of the Hamiltonian is . In terms of this overlap the spectral density function is
where is the Dirac delta function and is a dimensionless scaling parameter applied to the Hamiltonian, that we introduce through the algorithm itself.
Depending on the concrete implementation of SPE, we may never directly resolve the spectral density function exactly, but we construct a signal that will have features in common with it. This is done by first picking a convolution function such that when the spectral density is convolved by , the eigenvalues manifest as special points in the resulting signal
Here defines the support of , while determines where the eigenvalues are laid out in the resulting signal. We here skip the full derivation, but the interested reader is directed to our paper summarizing these ideas while providing benchmarks of SPE. Here it suffices to say that we can express this convolution in terms of the Fourier components as
Here is a parameter that gives the upper bound on the number of Fourier modes included in the algorithm. From this expression and the preceeding discussion we see that this signal can be recovered using a quantum processor. The expectation value of the time-evolution operator can be evaluated using the Hadamard test. The only other component we need is a Fourier transform of the convolution function. The convolution function itself can be chosen depending on desired algorithm scaling.
The convolution function is only approximately represented in the convolution integral because of the cut-off . This defines a trade-off between the accuracy of the convolution function and the run-time needed to accurately represent time-evolution by .
Finally, the scaling of the integrand with is not carried out by direct multiplication. Instead is sampled to determine the number of samples used to evaluate .
In the following we will show how to proceed with QURI Algo's SPE implementation with the LT22 variant is chosen in which is a Heaviside step-function.
The LT22 algorithm
We start with the LT22 variant. First we have to specify:
- : the maximum index of the Fourier modes considered,
- : a parameter that controls the interval within which the error of the Fourier transformed Heaviside step function is bounded.
- : Number of samples used to estimate the signal function .
These parameters specifies the step function used to convolute with the spectral density function and are wrapped inside StepFunctionParam
from quri_algo.algo.phase_estimation.spe import StepFunctionParam
d_max = 1000
delta = 1e-4
n_sample = 10000
signal_param = StepFunctionParam(d=d_max, delta=delta, n_sample=n_sample)
Next we normalize the spectrum by a factor of . It is a dimensionless factor that effectively scales first axis of the resulting signal, thus acting as a parameter for fine-graining the signal. The consequence of choosing a small is that the signal itself may not include the ground state energy or other eigenenergies of interest, however, choosing it too large will render the resolution too low to effectively distinguish different peaks in the signal. Here we pick .
from quri_algo.algo.phase_estimation.spe.lt22 import SingleSignalLT22GSEE
from pprint import pprint
tau = 1 / 20
lt22_algorithm = SingleSignalLT22GSEE(trotter_time_evolution_estimator, tau=tau)
The post processing part of the algorithm attempts to invert the signal function by finding the point where the height of the signal function takes the value of . Here, we choose to be 0.4. (One can rerun only the classical post processing part by changing )
eta = 0.4
spe_result = lt22_algorithm(state, signal_param, eta)
pprint(spe_result)
The algorithm returns a SPEResult
object which includes the estimated phase and the signal function , from which the phase is extracted.
lt22_gs_energy = spe_result.phase / tau
print(f"The obtained ground state energy is {lt22_gs_energy}")
print(f"The error is = {abs(lt22_gs_energy - np.min(vals))}")