^{1}, Ananth Grama

^{1}, Giorgos Kollias

^{1}and Sabre Kais

^{2,3,a)}

### Abstract

Unlike fixed designs, programmable circuit designs support an infinite number of operators. The functionality of a programmable circuit can be altered by simply changing the angle values of the rotation gates in the circuit. Here, we present a new quantum circuit design technique resulting in two general programmable circuit schemes. The circuit schemes can be used to simulate any given operator by setting the angle values in the circuit. This provides a fixed circuit design whose angles are determined from the elements of the given matrix–which can be non-unitary–in an efficient way. We also give both the classical and quantum complexity analysis for these circuits and show that the circuits require a few classical computations. For the electronic structure simulation on a quantum computer, one has to perform the following steps: prepare the initial wave function of the system; present the evolution operator *U* = *e* ^{−iHt } for a given atomic and molecular Hamiltonian *H* in terms of quantum gates array and apply the phase estimation algorithm to find the energy eigenvalues. Thus, in the circuit model of quantum computing for quantum chemistry, a crucial step is presenting the evolution operator for the atomic and molecular Hamiltonians in terms of quantum gate arrays. Since the presented circuit designs are independent from the matrix decomposition techniques and the global optimization processes used to find quantum circuits for a given operator, high accuracy simulations can be done for the unitary propagators of molecular Hamiltonians on quantum computers. As an example, we show how to build the circuit design for the hydrogen molecule.

This work is supported by the National Science Foundation (NSF) Centers for Chemical Innovation: Quantum Information for Quantum Chemistry, CHE-1037992.

I. INTRODUCTION

II. THE GENERAL SIMULATION IDEA

III. GENERATION OF PROGRAMMABLE CIRCUITS

A. The first circuit design

1. Formation step

2. Combination step

3. Input modification step

B. The second circuit design

1. Formation step

2. Combination step

3. Input modification

IV. COMPLEXITY ANALYSIS OF THE CIRCUITS

A. The complexity of the first circuit design

1. The classical complexity

2. The quantum complexity

B. The complexity of the second circuit

1. Classical complexity

2. The quantum complexity

C. Comparison with the non-programmable circuit designs

V. DISCUSSION AND CONCLUSION

A. Programmable quantum chips

B. Finding angles

C. Complex cases

D. Simulation of molecular Hamiltonians

### Key Topics

- Qubits
- 46.0
- Quantum computing
- 34.0
- Element formation
- 8.0
- Computer simulation
- 3.0
- Eigenvalues
- 2.0

## Figures

The number of qubits on the ancilla determines the number of *V* _{ i }s and hence the size of *V* in Eq. (2).

The number of qubits on the ancilla determines the number of *V* _{ i }s and hence the size of *V* in Eq. (2).

Block circuit diagram to simulate *U* by modifying the input |**0**⟩|ψ⟩ to and constructing *V* in two steps: the formation of the elements of *U* in *V* and bringing the same row elements in *U* to the first rows of *V* _{ i }s in *V*, combination. The necessary gates to form *V* and to also transform |**0**⟩|ψ⟩ to will generate the circuit.

Block circuit diagram to simulate *U* by modifying the input |**0**⟩|ψ⟩ to and constructing *V* in two steps: the formation of the elements of *U* in *V* and bringing the same row elements in *U* to the first rows of *V* _{ i }s in *V*, combination. The necessary gates to form *V* and to also transform |**0**⟩|ψ⟩ to will generate the circuit.

The first circuit design for a given general matrix: the initial Hadamards and the SWAPs are to modify the input, and the last Hadamards carry the elements to the first rows of *V* _{ i }s (combination step). The uniformly controlled quantum gates in the middle form all elements of *U* on the diagonal of *V* (formation step).

The first circuit design for a given general matrix: the initial Hadamards and the SWAPs are to modify the input, and the last Hadamards carry the elements to the first rows of *V* _{ i }s (combination step). The uniformly controlled quantum gates in the middle form all elements of *U* on the diagonal of *V* (formation step).

The second circuit with 4 × 4 initial blocks: The differently controlled quantum gates in the networks, after the *V* _{ i } blocks, combine small blocks and build the *N* × *N* blocks at the end. The initial Hadamards are for the modification of the input. The *V* _{ i } blocks are for the formation step.

The second circuit with 4 × 4 initial blocks: The differently controlled quantum gates in the networks, after the *V* _{ i } blocks, combine small blocks and build the *N* × *N* blocks at the end. The initial Hadamards are for the modification of the input. The *V* _{ i } blocks are for the formation step.

(a) A gray-coded multi-control network. (b) The decomposition of the gray-coded network in (a) into CNOT and single quantum gates.

(a) A gray-coded multi-control network. (b) The decomposition of the gray-coded network in (a) into CNOT and single quantum gates.

The circuit in (a) with 4 × 4 initial blocks can be represented as in (b) by using the circuit given in Fig. (7). Without changing the order of the gates having the same control state, the gates can be moved to form uniformly controlled networks as in (c): If a gate has the same angle value for all control states such as the control *X* gates in the circuit, they are equal to a single gate (in the case of *X* gates in the circuit, only one CNOT is required).

The circuit in (a) with 4 × 4 initial blocks can be represented as in (b) by using the circuit given in Fig. (7). Without changing the order of the gates having the same control state, the gates can be moved to form uniformly controlled networks as in (c): If a gate has the same angle value for all control states such as the control *X* gates in the circuit, they are equal to a single gate (in the case of *X* gates in the circuit, only one CNOT is required).

Quantum circuit which is found by following the Schmidt decomposition and can generate any vector of dimension 4 as the first row of its matrix representation.

Quantum circuit which is found by following the Schmidt decomposition and can generate any vector of dimension 4 as the first row of its matrix representation.

The circuit for the simulation of the hydrogen molecule. The angle values for the rotation gates are determined to create the elements of : There are only 19 rotation gates, the rest are *X* gates in order to get the right order for the elements after the combination. For diagonal elements of , these rotations are only around *z*-axis. For nonzero-diagonal elements, rotation about *z*-axis followed by rotations about *y*-axis. The angles for these gates are given in Table I.

The circuit for the simulation of the hydrogen molecule. The angle values for the rotation gates are determined to create the elements of : There are only 19 rotation gates, the rest are *X* gates in order to get the right order for the elements after the combination. For diagonal elements of , these rotations are only around *z*-axis. For nonzero-diagonal elements, rotation about *z*-axis followed by rotations about *y*-axis. The angles for these gates are given in Table I.

## Tables

Parameters for the rotation gates.

Parameters for the rotation gates.

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