Ansatz state for finding the lowest eigenvalue of a $2^n\times 2^n$ real matrix using VQE

Question

I would like to find the lowest eigenvalue of a $2^n\times 2^n$ real matrix $H$ using the VQE procedure. The measurement part is simple — I just expand $H$ in a sum of all possible $n$-qubit Pauli terms.

But I have not yet come up with a nice way of preparing an arbitrary $n$-qubit state with real amplitudes. What would be a good ansatz operator $U(\theta_1,\ldots\theta_{2^n-1})$ to do the job?

(If I were solving the problem in the second-quantized formalism using $2^n$ qubits, I would simply use the Unitary Coupled Cluster with $(2^n - 1)$ single amplitudes — does it have an analogue in such a 'binary' encoding??)

Answer 1

For preparing an arbitrary $n$-qubit state with real amplitudes, you can simply:

1. Take a circuit for preparing arbitrary states,
2. Replace all single-qubits rotations with $R_y$ rotations.

Here is how I do it in Qiskit:

def arbitrary_real_superposition_qiskit( amps: List[ float ], qc_arb = None ):
    _amps = np.array( amps )
    if not misc.is_power_of_two( _amps.shape[ 0 ] + 1 ):
        raise Exception( 'The number of parameters for arbitrary_real_superposition_qiskit() has to be 2^n-1.' )

    nqubit = int( np.log2( _amps.shape[ 0 ] + 1 ) )

    qc = qiskit.circuit.quantumcircuit.QuantumCircuit( nqubit )

    if type( qc_arb ) != qiskit.circuit.quantumcircuit.QuantumCircuit:
        qc_arb = init_state_qiskit( [ random.random() for i in range( _amps.shape[ 0 ] ) ] )

    i = 0
    for inst in qc_arb.data:
        if type( inst[ 0 ] ) == qiskit.circuit.library.standard_gates.u3.U3Gate \
                or type( inst[ 0 ] ) == qiskit.circuit.library.standard_gates.u2.U2Gate \
                or type( inst[ 0 ] ) == qiskit.circuit.library.standard_gates.u1.U1Gate:
            qc.ry( _amps[ i ], inst[ 1 ] )
            i = i + 1
        elif type( inst[ 0 ] ) == qiskit.circuit.library.standard_gates.x.CXGate:
            qc.cx( inst[ 1 ][ 0 ], inst[ 1 ][ 1 ] )
        else:
            raise Exception( 'A gate other than u1/u2/u3 and cx was encountered.' )
    return qc

where

def is_power_of_two( N ):
    """Returns True if n is a power of 2."""
    n = int(N)
    if n <= 0:
        return False
    else:
        return n & (n - 1) == 0 # How the hell is this even working??? :D

and

def init_state_qiskit( amps: List[ complex ] ):
 
    qubitn = int( np.log2( amps.shape[ 0 ] ) )

    qc = qiskit.circuit.quantumcircuit.QuantumCircuit( qubitn )
    qc.isometry( amps, list( range( qubitn ) ), None )
    
    while True:
        qc = qc.decompose()
        if qc == qc.decompose():
            break

    qc = qiskittranspile( qc, optimization_level = 3, basis_gates = [ 'u1', 'u2', 'u3', 'cx' ] )

    return qc

What happens here is that I initialize an $n$-qubit state from an arbitrary $2^n$ vector (to make sure that all the $2^n-1$ single-qubit rotations are present in the circuit), and replace those single-qubit rotations with $R_y$'s.

Answer 2

If you don't have any prior on your matrix, you can use the Strongly Entangling Circuit as a very generic ansatz for qubits.

Answer 3

Giving a general $2^n \times 2^n $ Hermitian matrix $H$, it's not guarantee that you can decompose/expand $H$ in polynomial strings of $n$-qubit Pauli matrices. So I don't know if VQE is a good method to find lowest eigenvalues of a matrix... unless you know that you can sum $H$ in polynomial terms of Pauli matrices. Even then, in general, VQE only gonna give you a good answer if you have some intuition about the problem. If you pick a variational form that is hardware efficient, using polynomial number of gates, then you can't expect it explore the entire Hilbert space...