vqls.py

"""Code for running VQLS on Rigetti's quantum computers."""

from itertools import product
from copy import deepcopy
from typing import (List, Tuple)

import numpy as np

import pyquil
from pyquil import Program, get_qc
import pyquil.gates as gates

# ====================================
# Pauli matrices and string dictionary
# ====================================

imat = np.array([[1, 0], [0, 1]], dtype=np.complex64)
xmat = np.array([[0, 1], [1, 0]], dtype=np.complex64)
ymat = np.array([[0, -1j], [1j, 0]], dtype=np.complex64)
zmat = np.array([[1, 0], [0, -1]], dtype=np.complex64)
pauli_dict = {"I": imat, "X": xmat, "Y": ymat, "Z": zmat}


def tensor(*matrices) -> np.ndarray:
    """Returns the tensor product of all matrices."""
    mats = list(matrices)
    mat = mats[0]
    if type(mat) != np.ndarray:
        raise ValueError("All arguments must be numpy.ndarray's.")
    for term in mats[1:]:
        mat = np.kron(mat, term)
    return mat


def matrix(coeffs: List[complex], terms: List[str]):
    """Returns the matrix defined by the coefficients and Paulis.
    
    Args:
        coeffs: List of real/complex numbers specifying coefficients of Paulis.
        terms: List of Pauli operators.
    """
    if len(coeffs) != len(terms):
        raise ValueError("Number of coeffs does not equal number of terms.")
    nqubits = len(terms[0])
    dim = 2**nqubits
    mat = np.zeros((dim, dim), dtype=np.complex64)
    for (c, pauli) in zip(coeffs, terms):
        pauli = [pauli_dict[key] for key in pauli]
        mat += c * tensor(*pauli)
    return mat


def matrix_of_hamiltonian(ham: List[Tuple[complex, str]]) -> np.ndarray:
    """Returns a matrix representation of the Hamiltonian in the standard basis."""
    coeffs = []
    paulis = []
    for (coeff, pauli) in ham:
        coeffs.append(coeff)
        paulis.append(pauli)
    mat = matrix(coeffs, paulis)
    return mat


def vector(coeffs: List[complex], terms: List[str]):
    """Returns the vector of the linear system.
    
    Args:
        coeffs: Weights of B matrix.
        terms: Pauli terms in B matrix.
    """
    return matrix(coeffs, terms)[:, 0]

# ======================================
# Functions for multipling Pauli strings
# ======================================

def mult(a: str, b: str) -> Tuple[complex, str]:
    """Returns product a times b of two single qubit Paulis."""
    if a not in ("I", "X", "Y", "Z") or b not in ("I", "X", "Y", "Z"):
        raise ValueError("One or more invalid Pauli keys.")
        
    prod = {"XY": "Z",
            "YZ": "X",
            "ZX": "Y"}

    if a == "I":
        return (1., b)
    if b == "I":
        return (1., a)
    if a == b:
        return (1., "I")
    
    phase = 1j
    new = a + b
    if new in prod.keys():
        return (phase, prod[new])
    else:
        return (-phase, prod[b + a])
    

def multn(a: str, b: str) -> Tuple[complex, str]:
    """Returns product a times b of two n qubit Paulis."""
    if len(a) != len(b):
        raise ValueError("len(a) != len(b)")
    phase = 1
    paulis = []
    for (aterm, bterm) in zip(a, b):
        p, pauli = mult(aterm, bterm)
        phase *= p
        paulis.append(pauli)
    return phase, "".join(paulis)

# ===========================================================
# Functions for expanding input Pauli string for Pb projector
# ===========================================================

def Pb_expansion(Bcoeffs: List[complex], Bterms: List[str]) -> List[Tuple[complex, str]]:
    """Inputs B as a linear combination of Paulis and 
    outputs B|0><0|B^dagger as a linear combination of Paulis.
    """
    if len(Bcoeffs) != len(Bterms):
        raise ValueError("len(Bcoeffs) != len(Bterms)")
    n = len(Bterms[0])
    
    # Form the |0><0| projector on n qubits
    terms = [["I", "Z"]] * n
    prod = list(product(*terms))
    Pterms = ["".join(p) for p in prod]
    
    coeffs = []
    paulis = []
    
    for ii in range(len(Bterms)):
        for jj in range(len(Pterms)):
            for kk in range(len(Bterms)):
                coeff = Bcoeffs[ii] * np.conj(Bcoeffs[kk]) / 2**n
                phase1, pauli1 = multn(Bterms[ii], Pterms[jj])
                phase2, pauli = multn(pauli1, Bterms[kk])
                
                coeffs.append(coeff * phase1 * phase2)
                paulis.append(pauli)
    return coeffs, paulis

# ====================================================
# Functions to attempt to minimize the number of terms 
# in the Pauli expansion of the Hamiltonian.
# ====================================================

def combine_paulis(
    hamiltonian: List[Tuple[complex, str]]
) -> List[Tuple[complex, str]]:
    """Combines identical Pauli terms in the Hamiltonian, if possible."""
    new = [list(hamiltonian[0])]
    seen_paulis = [new[0][1]]

    for (coeff, pauli) in hamiltonian[1:]:
        if pauli in seen_paulis:
            # Find the index of where the Pauli is in new
            index = seen_paulis.index(pauli)
            new[index][0] += coeff
        else:
            new.append(list((coeff, pauli)))
            seen_paulis.append(pauli)

    return new


def drop_zero(
    hamiltonian: List[Tuple[complex, str]]
) -> List[Tuple[complex, str]]:
    """Drops Pauli terms with zero weight."""
    new = []
    for (coeff, pauli) in hamiltonian:
        if not np.isclose(abs(coeff), 0.0):
            new.append([coeff, pauli])
    return new

# =================================================================
# Function for getting the effective Hamiltonian of a linear system
# =================================================================

def effective_hamiltonian(Acoeffs: List[complex], 
                          Aterms: List[str], 
                          Bcoeffs: List[complex],
                          Bterms: List[str]) -> Tuple[List[complex], List[str]]:
    """Returns the effective Hamiltonian of an input linear system Ax = B|0>.
    
    Args:
        Acoeffs: List of coefficients of A matrix.
        Aterms: Pauli strings for each coefficient of A matrix.
        Bcoeffs: List of coefficients of B matrix.
        Bterms: Pauli strings for each coefficient of B matrix.
    """
    # Input checks
    if len(Acoeffs) != len(Aterms):
        raise ValueError("len(Acoeffs) != len(Aterms)")
    if len(Bcoeffs) != len(Bterms):
        raise ValueError("len(Bceoffs) != len(Bterms)")

    if not np.isclose(np.linalg.norm(vector(Bcoeffs, Bterms)), 1.0):
        raise ValueError("The bvector must have unit norm.")

    # Initialize the Hamiltonian as a List[Tuple[complex, str]]
    ham = []
    n = len(Aterms[0])
    
    # =======================================
    # First term in Hamiltonian (A^\dagger A)
    # =======================================

    for l in range(len(Aterms)):
        for k in range(len(Aterms)):
            coeff = np.conj(Acoeffs[l]) * Acoeffs[k]
            phase, pauli = multn(Aterms[l], Aterms[k])
            ham.append(list((coeff * phase, pauli)))
    
    ham = drop_zero(combine_paulis(ham))
    
    # ============================================
    # Second term in Hamiltonian (A^\dagger P_b A)
    # ============================================
    
    # Get the Pb expansion
    Pbcoeffs, Pbterms = Pb_expansion(Bcoeffs, Bterms)
    
    for k in range(len(Aterms)):
        for l in range(len(Aterms)):
            for m in range(len(Pbterms)):
                # Compute this coefficient and term
                coeff = np.conj(Acoeffs[k]) * Acoeffs[l] * Pbcoeffs[m]
                phase1, pauli1 = multn(Aterms[k], Pbterms[m])
                phase2, pauli = multn(pauli1, Aterms[l])
                
                # Append it to the expansion
                ham.append(list((-1 * phase1 * phase2 * coeff, pauli)))
    
    return drop_zero(combine_paulis(ham))


# ==================================================
# Functions for computing <H> with a quantum circuit
# ==================================================

def yansatz(computer):
    """Returns a circuit with a product state ansatz."""
    n = len(computer.qubits())
    # Get a circuit and classical memory register
    circ = Program()
    creg = circ.declare("ro", memory_type="BIT", memory_size=n)

    # Define parameters for the ansatz
    angles = circ.declare("theta", memory_type="REAL", memory_size=n)

    # Add the ansatz
    circ += [gates.RY(angles[ii], computer.qubits()[ii]) for ii in range(n)]
    
    return circ, creg


def yansatzCZ(computer):
    """Returns a Ry ansatz with some entanglement."""
    # Grab the qubits
    qubits = tuple(computer.qubits())
    n = len(qubits)
    
    # Get a circuit and classical memory register
    circ = Program()
    creg = circ.declare("ro", memory_type="BIT", memory_size=n)

    # Define parameters for the ansatz
    angles = circ.declare("theta", memory_type="REAL", memory_size=2 * n)

    # Add the ansatz
    circ += [gates.RY(angles[ii], qubits[ii]) for ii in range(n)]
    circ += [gates.CZ(qubits[ii], qubits[ii + 1]) for ii in range(n - 1)]
    circ += [gates.RZ(angles[ii], qubits[ii]) for ii in range(n)]
    circ += [gates.CZ(qubits[-ii - 1], qubits[-ii - 2]) for ii in range(n - 1)]
    
    return circ, creg


def yansatzCZrotations(computer):
    """Returns a Ry ansatz with some entanglement."""
    # Grab the qubits
    qubits = tuple(computer.qubits())
    n = len(qubits)
    
    # Get a circuit and classical memory register
    circ = Program()
    creg = circ.declare("ro", memory_type="BIT", memory_size=n)

    # Define parameters for the ansatz
    angles = circ.declare("theta", memory_type="REAL", memory_size=3 * n)

    # Add the ansatz
    circ += [gates.RY(angles[ii], qubits[ii]) for ii in range(n)]
    circ += [gates.CZ(qubits[ii], qubits[ii + 1]) for ii in range(n - 1)]
    circ += [gates.RZ(angles[ii], qubits[ii]) for ii in range(n)]
    circ += [gates.CZ(qubits[ii + 1], qubits[ii + 2]) for ii in range(n - 2)]
    circ += [gates.RY(angles[ii], qubits[ii]) for ii in range(n)]
    circ += [gates.CZ(qubits[ii], qubits[ii + 1]) for ii in range(n - 1)]
    
    return circ, creg


def expectation(angles: List[float], 
                coeff: complex, 
                pauli: str,
                ansatz: pyquil.Program,
                creg: pyquil.quilatom.MemoryReference,
                computer: pyquil.api.QuantumComputer,
                shots: int = 10000,
                verbose: bool = False) -> float:
    """Returns coeff * <\theta| paulii |\theta>.
    
    Args:
        angles: List of angles at which to evaluate coeff * <theta| pauli |theta>.
        coeff: Coefficient of Pauli term.
        pauli: Pauli string.
        ansatz: pyQuil program representing the ansatz state.
        creg: Classical register of ansatz to measure into.
        computer: QuantumComputer to execute the circuit on.
        shots: Number of times to execute the circuit (sampling statistics).
        verbose: Option for visualization/debugging.
    """
    if np.isclose(np.imag(coeff), 0.0):
        coeff = np.real(coeff)
    
    if set(pauli) == {"I"}:
        return coeff
        
    angles = list(angles)
    angles = deepcopy(angles)
    
    if verbose:
        print("DEBUG holy fuck")
        print(f"type(angles) = {type(angles)}")
        print("angles =", angles)
    
    # Set up the circuit
    circuit = ansatz.copy()
    qubits = computer.qubits()
    measured = []
    for (q, p) in enumerate(pauli):
        if p in ("X", "Y", "Z"):
            measured.append(qubits[q])
        if p == "X":
            circuit += [gates.H(qubits[q]), gates.MEASURE(qubits[q], creg[q])]
        elif p == "Y":
            circuit += [gates.S(qubits[q]), gates.H(qubits[q]), gates.MEASURE(qubits[q], creg[q])]
        elif p == "Z":
            circuit += [gates.MEASURE(qubits[q], creg[q])]
    
    if verbose:
        print(f"Computing {coeff} x <theta|{pauli}|theta>...")
        print("\nCircuit to be executed:")
        print(circuit)
        print(f"type(angles) = {type(angles)}")
    
    # Execute the circuit
    circuit.wrap_in_numshots_loop(shots)
    executable = computer.compile(circuit)
    res = computer.run(executable, memory_map={"theta": angles})
    
    if verbose:
        print("\nResults:")
        print(f"{len(res)} total measured bit strings.")
        print(res)
    
    # Do the postprocessing
    tot = 0.0
    for vals in res:
        tot += (-1)**sum(vals)
    return coeff * tot / shots


# TODO: Utilize simulatenous measurements
def energy(angles, hamiltonian, ansatz, creg, computer, shots=10000, min_weight=0, verbose=False):
    """Returns <theta| H |theta>.
    
    Args:
        angles: List of angles at which to evaluate <theta| H |theta>.
        hamiltonian: List[Tuple] of (coeff, pauli) pairs.
        ansatz: pyQuil program representing the ansatz state.
        creg: Classical register of ansatz to measure into.
        computer: QuantumComputer to execute the circuit on.
        shots: Number of times to execute the circuit (sampling statistics).
        min_weight: If a term has abs(coeff) < min_weight, skip the term.
                    Default value = 0, i.e., all terms are present in computation.
        verbose: Option for visualization/debugging.
    """
    value = 0.0
    for (coeff, pauli) in hamiltonian:
        if abs(coeff) >= min_weight:
            value += expectation(angles, coeff, pauli, ansatz, creg, computer, shots, verbose)
    return value

# ==========================================================================
# Function for getting the wavefunction from the ansatz + optimal parameters
# ==========================================================================

def qsolution(ansatz, opt_angles):
    """Returns the wavefunction of the ansatz at the optimal angles."""
    prog = Program()
    memory_map = {"theta": opt_angles}
    for name, arr in memory_map.items():
        for index, value in enumerate(arr):
            prog += gates.MOVE(gates.MemoryReference(name, offset=index), value)

    ansatz = prog + ansatz
    soln = pyquil.quil.percolate_declares(ansatz)

    wfsim = pyquil.api.WavefunctionSimulator()
    return wfsim.wavefunction(soln).amplitudes


# =======================================
# Functions for simultaneous measurements
# =======================================

def is_sim_meas(op1: str, op2: str) -> bool:
    """Returns True if op1 and op2 can be simultaneously measured.
    
    Args:
        op1: Pauli string on n qubits.
        op2: Pauli string on n qubits.
    
    Examples:
        is_sim_meas("IZI", "XIX") -> True
        
        is_sim_meas("XZ", "XX") -> False
    """
    if len(op1) != len(op2):
        raise ValueError(
            "Pauli operators act on different numbers of qubits."
        )
    
    for (a, b) in zip(op1, op2):
        if a != b and a != "I" and b != "I":
            return False
    return True


def can_be_grouped_with(op: str, group: List[str]) -> bool:
    """Returns True if op can be simultaneously measured with every
    operator in the given group, else False.
    
    Args:
        op: A Pauli operator.
        group: A list of Pauli operators.
    """
    for other in group:
        if not is_sim_meas(op, other):
            return False
    return True


def is_sim_meas_group(group: List[str]) -> bool:
    """Returns True if every operator within the set is pairwise simultaneously measurable."""
    for ii in range(len(group) - 1):
        for jj in range(ii + 1, len(group)):
            if not is_sim_meas(group[ii], group[jj]):
                return False
    return True


def split_ham_to_coeffs_and_paulis(ham):
    """Returns list of coeffs and paulis from input hamiltonian."""
    ham = np.array(ham)
    coeffs = list(map(lambda x: float(x), ham[:, 0]))
    paulis = list(ham[:, 1])

    return coeffs, paulis


def group_greedy(ham, randomized: bool = True):
    """Groups the hamiltonian terms into simultaneously measurable sets using a greedy approach."""
    # Shuffle the terms in the Hamiltonian, if desired
    if randomized:
        np.random.shuffle(ham)
    
    # Split into coeffs and Paulis for convenience
    coeffs, paulis = split_ham_to_coeffs_and_paulis(ham)
    
    # Do the grouping
    groups = [[paulis.pop(0)]]
    cgroups = [[coeffs.pop(0)]]

    maxit = len(paulis)
    it = 0
    while paulis:
        added = False
        for (ii, group) in enumerate(groups):
            if can_be_grouped_with(paulis[0], group):
                groups[ii].append(paulis.pop(0))
                cgroups[ii].append(coeffs.pop(0))
                added = True
                break
        if not added and len(paulis) > 0:
            groups.append([paulis.pop(0)])
            cgroups.append([coeffs.pop(0)])

        it += 1
        if it > maxit:
            raise SystemError("Internal error with group_greedy algorithm, max iterations exceeded.")

    if not (coeffs == [] and paulis == []):
        print(coeffs)
        print(paulis)
        raise SystemError("Internal error with group_greedy algorithm.")

    # Recombine into Hamiltonian
    ham = []
    for cgroup, group in zip(cgroups, groups):
        ham.append(
            list(zip(cgroup, group))
        )
    return ham


def measure_group(
    angles: List[float],
    group,
    ansatz: pyquil.Program,
    creg: pyquil.quilatom.MemoryReference,
    computer: pyquil.api.QuantumComputer,
    shots: int = 10_000,
    verbose: bool = False
) -> float:
    """Returns the expectation over all Pauli operators in the group by
    executing a single circuit.
    
    Args:
        angles: List of angles at which to evaluate coeff * <theta| pauli |theta>.
        group: Group of simultaneously measurable Pauli operators.
               Format:
                   [(coeff1, pauli1),
                    (coeff2, pauli2),
                    ...
                    (coeffn, paulin)]
        ansatz: pyQuil program representing the ansatz state.
        creg: Classical register of ansatz to measure into.
        computer: QuantumComputer to execute the circuit on.
        shots: Number of times to execute the circuit (sampling statistics).
    """
    angles = list(angles)
    angles = deepcopy(angles)
    
    # Split the group into coefficients and paulis
    coeffs, paulis = split_ham_to_coeffs_and_paulis(group)

    if not is_sim_meas_group(paulis):
        raise ValueError("Input group is not simultaneously measurable.")
    
    # Squash the group into one Pauli string to determine the correct measurements
    #  Note this is only possible by assumption that the group is simultaneously measurable
    squashed = squash(paulis)
    
    # Add the right rotation + measurement operators to the ansatz
    circuit = ansatz.copy()
    qubits = computer.qubits()
    to_measure = []
    for (q, p) in enumerate(squashed):
        if p in ("X", "Y", "Z"):
            to_measure.append(q)
        if p == "X":
            circuit += [gates.H(qubits[q])]
        elif p == "Y":
            circuit += [gates.S(qubits[q]), gates.H(qubits[q])]
    
    # Add the terminal measurements
    #  Note we do it this way since all measurements *must* be at the end of the circuit on hardware
    for q in to_measure:
        circuit += [gates.MEASURE(qubits[q], creg[q])]
            
    # Execute the circuit
    circuit.wrap_in_numshots_loop(shots)
    executable = computer.compile(circuit)
    res = computer.run(executable, memory_map={"theta": angles})
    
    # Do the postprocessing
    supports = [support(pauli) for pauli in paulis]
    total = 0.0
    for ii in range(len(supports)):
        tot = 0.0
        for vals in res:
            sliced = islice(vals, supports[ii])
            tot += (-1)**sum(sliced)
        total += coeffs[ii] * tot / shots
    return total


def energy_sim(
    angles: List[float],
    ham,
    ansatz: pyquil.Program,
    creg: pyquil.quilatom.MemoryReference,
    computer: pyquil.api.QuantumComputer,
    shots: int = 10_000
) -> float:
    """Returns <H> using simultaneous measurements.
    
    Args:
        angles: List of angles at which to evaluate coeff * <theta| pauli |theta>.
        ham: List of groups of simultaneously measurable Pauli operators.
               Format: ham = [group1, group2, ..., groupM] where each group has the form
                   [(coeff1, pauli1),
                    (coeff2, pauli2),
                    ...
                    (coeffn, paulin)]
        REQUIRES: Each group to be simultaneously measurable.
        ansatz: pyQuil program representing the ansatz state.
        creg: Classical register of ansatz to measure into.
        computer: QuantumComputer to execute the circuit on.
        shots: Number of times to execute the circuit (sampling statistics).
    """
    tot = 0.0
    for group in ham:
        tot += measure_group(angles, group, ansatz, creg, computer, shots)
    return tot


def char(cA, cB):
    """Returns the appropriate single qubit pauli character when merging."""
    if cA == "I":
        return cB
    return cA


def merge(opA: str, opB: str) -> str:
    """Merges two pauli operators into a single squashed operator."""
    new = ""
    for (cA, cB) in zip(opA, opB):
        new += char(cA, cB)
    return new


def squash(group):
    """Merges a group of n puali operators into a single squashed operator."""
    squashed = group[0]
    for pauli in group[1:]:
        squashed = merge(squashed, pauli)
    return squashed


def support(pauli: str):
    """Returns indices where the Pauli string is non-identity."""
    has_support = lambda c: c != "I"
    inds = []
    for (i, p) in enumerate(pauli):
        if has_support(p):
            inds.append(i)
    return inds


def islice(vals, inds):
    """Returns a list of values from `vals` at the indices in `inds`."""
    sliced = []
    for (i, v) in enumerate(vals):
        if i in inds:
            sliced.append(v)
    return sliced