Circuit Synthesis

qiskit.synthesis

Evolution Synthesis

Linear Function Synthesis

qiskit.synthesis.synth_cnot_count_full_pmh(state, section_size=2)

Synthesize linear reversible circuits for all-to-all architecture using Patel, Markov and Hayes method.

This function is an implementation of the Patel, Markov and Hayes algorithm from [1] for optimal synthesis of linear reversible circuits for all-to-all architecture, as specified by an n x n matrix.

Parameters

• state (list (opens in a new tab)[list (opens in a new tab)] or ndarray) – n x n boolean invertible matrix, describing the state of the input circuit
• section_size (int (opens in a new tab)) – the size of each section, used in the Patel–Markov–Hayes algorithm [1]. section_size must be a factor of num_qubits.

Returns

a CX-only circuit implementing the linear transformation.

Return type

QuantumCircuit

Raises

QiskitError – when variable “state” isn’t of type numpy.ndarray

References

1. Patel, Ketan N., Igor L. Markov, and John P. Hayes, Optimal synthesis of linear reversible circuits, Quantum Information & Computation 8.3 (2008): 282-294. arXiv:quant-ph/0302002 [quant-ph] (opens in a new tab)

qiskit.synthesis.synth_cnot_depth_line_kms(mat)

Synthesize linear reversible circuit for linear nearest-neighbor architectures using Kutin, Moulton, Smithline method.

Synthesis algorithm for linear reversible circuits from [1], Chapter 7. Synthesizes any linear reversible circuit of n qubits over linear nearest-neighbor architecture using CX gates with depth at most 5*n.

Parameters

mat (np.ndarray]) – A boolean invertible matrix.

Returns

the synthesized quantum circuit.

Return type

QuantumCircuit

Raises

QiskitError – if mat is not invertible.

References

1. Kutin, S., Moulton, D. P., Smithline, L., Computation at a distance, Chicago J. Theor. Comput. Sci., vol. 2007, (2007), arXiv:quant-ph/0701194 (opens in a new tab)

Linear-Phase Synthesis

qiskit.synthesis.synth_cz_depth_line_mr(mat)

Synthesis of a CZ circuit for linear nearest neighbour (LNN) connectivity, based on Maslov and Roetteler.

Note that this method reverts the order of qubits in the circuit, and returns a circuit containing CX and phase (S, Sdg or Z) gates.

Parameters

mat (ndarray (opens in a new tab)) – an upper-diagonal matrix representing the CZ circuit. mat[i][j]=1 for i<j represents a CZ(i,j) gate

Returns

a circuit implementation of the CZ circuit of depth 2*n+2 for LNN connectivity.

Return type

QuantumCircuit

Reference:

1. Dmitri Maslov, Martin Roetteler, Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations, arXiv:1705.09176 (opens in a new tab).

qiskit.synthesis.synth_cx_cz_depth_line_my(mat_x, mat_z)

Joint synthesis of a -CZ-CX- circuit for linear nearest neighbour (LNN) connectivity, with 2-qubit depth at most 5n, based on Maslov and Yang. This method computes the CZ circuit inside the CX circuit via phase gate insertions.

Parameters

• mat_z (ndarray (opens in a new tab)) – a boolean symmetric matrix representing a CZ circuit. Mz[i][j]=1 represents a CZ(i,j) gate
• mat_x (ndarray (opens in a new tab)) – a boolean invertible matrix representing a CX circuit.

Returns

a circuit implementation of a CX circuit following a CZ circuit, denoted as a -CZ-CX- circuit,in two-qubit depth at most 5n, for LNN connectivity.

Return type

QuantumCircuit

Reference:

1. Kutin, S., Moulton, D. P., Smithline, L., Computation at a distance, Chicago J. Theor. Comput. Sci., vol. 2007, (2007), arXiv:quant-ph/0701194 (opens in a new tab)
2. Dmitri Maslov, Willers Yang, CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate, arXiv:2210.16195 (opens in a new tab).

Permutation Synthesis

qiskit.synthesis.synth_permutation_depth_lnn_kms(pattern)

Synthesize a permutation circuit for a linear nearest-neighbor architecture using the Kutin, Moulton, Smithline method.

This is the permutation synthesis algorithm from https://arxiv.org/abs/quant-ph/0701194 (opens in a new tab), Chapter 6. It synthesizes any permutation of n qubits over linear nearest-neighbor architecture using SWAP gates with depth at most n and size at most n(n-1)/2 (where both depth and size are measured with respect to SWAPs).

Parameters

pattern (Union[list (opens in a new tab)[int (opens in a new tab)], np.ndarray]) – permutation pattern, describing which qubits occupy the positions 0, 1, 2, etc. after applying the permutation. That is, pattern[k] = m when the permutation maps qubit m to position k. As an example, the pattern [2, 4, 3, 0, 1] means that qubit 2 goes to position 0, qubit 4 goes to position 1, etc.

Returns

the synthesized quantum circuit.

Return type

QuantumCircuit

qiskit.synthesis.synth_permutation_basic(pattern)

Synthesize a permutation circuit for a fully-connected architecture using sorting.

More precisely, if the input permutation is a cycle of length m, then this creates a quantum circuit with m-1 SWAPs (and of depth m-1); if the input permutation consists of several disjoint cycles, then each cycle is essentially treated independently.

Parameters

pattern (Union[list (opens in a new tab)[int (opens in a new tab)], np.ndarray]) – permutation pattern, describing which qubits occupy the positions 0, 1, 2, etc. after applying the permutation. That is, pattern[k] = m when the permutation maps qubit m to position k. As an example, the pattern [2, 4, 3, 0, 1] means that qubit 2 goes to position 0, qubit 4 goes to position 1, etc.

Returns

the synthesized quantum circuit.

Return type

QuantumCircuit

qiskit.synthesis.synth_permutation_acg(pattern)

Synthesize a permutation circuit for a fully-connected architecture using the Alon, Chung, Graham method.

This produces a quantum circuit of depth 2 (measured in the number of SWAPs).

This implementation is based on the Theorem 2 in the paper “Routing Permutations on Graphs Via Matchings” (1993), available at https://www.cs.tau.ac.il/~nogaa/PDFS/r.pdf (opens in a new tab).

Parameters

pattern (Union[list (opens in a new tab)[int (opens in a new tab)], np.ndarray]) – permutation pattern, describing which qubits occupy the positions 0, 1, 2, etc. after applying the permutation. That is, pattern[k] = m when the permutation maps qubit m to position k. As an example, the pattern [2, 4, 3, 0, 1] means that qubit 2 goes to position 0, qubit 4 goes to position 1, etc.

Returns

the synthesized quantum circuit.

Return type

QuantumCircuit

Clifford Synthesis

qiskit.synthesis.synth_clifford_full(clifford, method=None)

Decompose a Clifford operator into a QuantumCircuit.

For N <= 3 qubits this is based on optimal CX cost decomposition from reference [1]. For N > 3 qubits this is done using the general non-optimal greedy compilation routine from reference [3], which typically yields better CX cost compared to the AG method in [2].

Parameters

• clifford (Clifford) – a clifford operator.
• method (str (opens in a new tab)) – Optional, a synthesis method (‘AG’ or ‘greedy’). If set this overrides optimal decomposition for N <=3 qubits.

Returns

a circuit implementation of the Clifford.

Return type

QuantumCircuit

References

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph] (opens in a new tab)
2. S. Aaronson, D. Gottesman, Improved Simulation of Stabilizer Circuits, Phys. Rev. A 70, 052328 (2004). arXiv:quant-ph/0406196 (opens in a new tab)
3. Sergey Bravyi, Shaohan Hu, Dmitri Maslov, Ruslan Shaydulin, Clifford Circuit Optimization with Templates and Symbolic Pauli Gates, arXiv:2105.02291 [quant-ph] (opens in a new tab)

qiskit.synthesis.synth_clifford_ag(clifford)

Decompose a Clifford operator into a QuantumCircuit based on Aaronson-Gottesman method.

Parameters

clifford (Clifford) – a clifford operator.

Returns

a circuit implementation of the Clifford.

Return type

QuantumCircuit

Reference:

1. S. Aaronson, D. Gottesman, Improved Simulation of Stabilizer Circuits, Phys. Rev. A 70, 052328 (2004). arXiv:quant-ph/0406196 (opens in a new tab)

qiskit.synthesis.synth_clifford_bm(clifford)

Optimal CX-cost decomposition of a Clifford operator on 2-qubits or 3-qubits into a QuantumCircuit based on Bravyi-Maslov method.

Parameters

clifford (Clifford) – a clifford operator.

Returns

a circuit implementation of the Clifford.

Return type

QuantumCircuit

Raises

QiskitError – if clifford is on more than 3 qubits.

Reference:

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph] (opens in a new tab)

qiskit.synthesis.synth_clifford_greedy(clifford)

Decompose a Clifford operator into a QuantumCircuit based on the greedy Clifford compiler that is described in Appendix A of Bravyi, Hu, Maslov and Shaydulin.

This method typically yields better CX cost compared to the Aaronson-Gottesman method.

Parameters

clifford (Clifford) – a clifford operator.

Returns

a circuit implementation of the Clifford.

Return type

QuantumCircuit

Raises

QiskitError – if symplectic Gaussian elimination fails.

Reference:

1. Sergey Bravyi, Shaohan Hu, Dmitri Maslov, Ruslan Shaydulin, Clifford Circuit Optimization with Templates and Symbolic Pauli Gates, arXiv:2105.02291 [quant-ph] (opens in a new tab)

qiskit.synthesis.synth_clifford_layers(cliff, cx_synth_func=<function _default_cx_synth_func>, cz_synth_func=<function _default_cz_synth_func>, cx_cz_synth_func=None, cz_func_reverse_qubits=False, validate=False)

Synthesis of a Clifford into layers, it provides a similar decomposition to the synthesis described in Lemma 8 of Bravyi and Maslov.

For example, a 5-qubit Clifford circuit is decomposed into the following layers:

     ┌─────┐┌─────┐┌────────┐┌─────┐┌─────┐┌─────┐┌─────┐┌────────┐
q_0: ┤0    ├┤0    ├┤0       ├┤0    ├┤0    ├┤0    ├┤0    ├┤0       ├
     │     ││     ││        ││     ││     ││     ││     ││        │
q_1: ┤1    ├┤1    ├┤1       ├┤1    ├┤1    ├┤1    ├┤1    ├┤1       ├
     │     ││     ││        ││     ││     ││     ││     ││        │
q_2: ┤2 S2 ├┤2 CZ ├┤2 CX_dg ├┤2 H2 ├┤2 S1 ├┤2 CZ ├┤2 H1 ├┤2 Pauli ├
     │     ││     ││        ││     ││     ││     ││     ││        │
q_3: ┤3    ├┤3    ├┤3       ├┤3    ├┤3    ├┤3    ├┤3    ├┤3       ├
     │     ││     ││        ││     ││     ││     ││     ││        │
q_4: ┤4    ├┤4    ├┤4       ├┤4    ├┤4    ├┤4    ├┤4    ├┤4       ├
     └─────┘└─────┘└────────┘└─────┘└─────┘└─────┘└─────┘└────────┘

This decomposition is for the default cz_synth_func and cx_synth_func functions, with other functions one may see slightly different decomposition.

Parameters

• cliff (Clifford) – a clifford operator.
• cx_synth_func (Callable) – a function to decompose the CX sub-circuit. It gets as input a boolean invertible matrix, and outputs a QuantumCircuit.
• cz_synth_func (Callable) – a function to decompose the CZ sub-circuit. It gets as input a boolean symmetric matrix, and outputs a QuantumCircuit.
• cx_cz_synth_func (Callable) – optional, a function to decompose both sub-circuits CZ and CX.
• validate (Boolean) – if True, validates the synthesis process.
• cz_func_reverse_qubits (Boolean) – True only if cz_synth_func is synth_cz_depth_line_mr, since this function returns a circuit that reverts the order of qubits.

Returns

a circuit implementation of the Clifford.

Return type

QuantumCircuit

Reference:

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph] (opens in a new tab)

qiskit.synthesis.synth_clifford_depth_lnn(cliff)

Synthesis of a Clifford into layers for linear-nearest neighbour connectivity.

The depth of the synthesized n-qubit circuit is bounded by 7*n+2, which is not optimal. It should be replaced by a better algorithm that provides depth bounded by 7*n-4 [3].

Parameters

cliff (Clifford) – a clifford operator.

Returns

a circuit implementation of the Clifford.

Return type

QuantumCircuit

Reference:

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph] (opens in a new tab)
2. Dmitri Maslov, Martin Roetteler, Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations, arXiv:1705.09176 (opens in a new tab).
3. Dmitri Maslov, Willers Yang, CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate, arXiv:2210.16195 (opens in a new tab).

CNOTDihedral Synthesis

qiskit.synthesis.synth_cnotdihedral_full(elem)

Decompose a CNOTDihedral element into a QuantumCircuit. For N <= 2 qubits this is based on optimal CX cost decomposition from reference [1]. For N > 2 qubits this is done using the general non-optimal compilation routine from reference [2].

Parameters

elem (CNOTDihedral) – a CNOTDihedral element.

Returns

a circuit implementation of the CNOTDihedral element.

Return type

QuantumCircuit

References

1. Shelly Garion and Andrew W. Cross, Synthesis of CNOT-Dihedral circuits with optimal number of two qubit gates, Quantum 4(369), 2020 (opens in a new tab)
2. Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin and Jay M. Gambetta, Scalable randomised benchmarking of non-Clifford gates, npj Quantum Inf 2, 16012 (2016).

qiskit.synthesis.synth_cnotdihedral_two_qubits(elem)

Decompose a CNOTDihedral element on a single qubit and two qubits into a QuantumCircuit. This decomposition has an optimal number of CX gates.

Parameters

elem (CNOTDihedral) – a CNOTDihedral element.

Returns

a circuit implementation of the CNOTDihedral element.

Return type

QuantumCircuit

Raises

QiskitError – if the element in not 1-qubit or 2-qubit CNOTDihedral.

Reference:

1. Shelly Garion and Andrew W. Cross, On the structure of the CNOT-Dihedral group, arXiv:2006.12042 [quant-ph] (opens in a new tab)

qiskit.synthesis.synth_cnotdihedral_general(elem)

Decompose a CNOTDihedral element into a QuantumCircuit.

Decompose a general CNOTDihedral elements. The number of CNOT gates is not necessarily optimal. For a decomposition of a 1-qubit or 2-qubit element, call synth_cnotdihedral_two_qubits.

Parameters

elem (CNOTDihedral) – a CNOTDihedral element.

Returns

a circuit implementation of the CNOTDihedral element.

Return type

QuantumCircuit

Raises

QiskitError – if the element could not be decomposed into a circuit.

Reference:

1. Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin and Jay M. Gambetta, Scalable randomised benchmarking of non-Clifford gates, npj Quantum Inf 2, 16012 (2016).

Stabilizer State Synthesis

qiskit.synthesis.synth_stabilizer_layers(stab, cz_synth_func=<function _default_cz_synth_func>, cz_func_reverse_qubits=False, validate=False)

Synthesis of a stabilizer state into layers.

It provides a similar decomposition to the synthesis described in Lemma 8 of Bravyi and Maslov, without the initial Hadamard-free sub-circuit which do not affect the stabilizer state.

For example, a 5-qubit stabilizer state is decomposed into the following layers:

     ┌─────┐┌─────┐┌─────┐┌─────┐┌────────┐
q_0: ┤0    ├┤0    ├┤0    ├┤0    ├┤0       ├
     │     ││     ││     ││     ││        │
q_1: ┤1    ├┤1    ├┤1    ├┤1    ├┤1       ├
     │     ││     ││     ││     ││        │
q_2: ┤2 H2 ├┤2 S1 ├┤2 CZ ├┤2 H1 ├┤2 Pauli ├
     │     ││     ││     ││     ││        │
q_3: ┤3    ├┤3    ├┤3    ├┤3    ├┤3       ├
     │     ││     ││     ││     ││        │
q_4: ┤4    ├┤4    ├┤4    ├┤4    ├┤4       ├
     └─────┘└─────┘└─────┘└─────┘└────────┘

Parameters

• stab (StabilizerState) – a stabilizer state.
• cz_synth_func (Callable) – a function to decompose the CZ sub-circuit. It gets as input a boolean symmetric matrix, and outputs a QuantumCircuit.
• validate (Boolean) – if True, validates the synthesis process.
• cz_func_reverse_qubits (Boolean) – True only if cz_synth_func is synth_cz_depth_line_mr, since this function returns a circuit that reverts the order of qubits.

Returns

a circuit implementation of the stabilizer state.

Return type

QuantumCircuit

Raises

QiskitError – if the input is not a StabilizerState.

Reference:

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph] (opens in a new tab)

qiskit.synthesis.synth_stabilizer_depth_lnn(stab)

Synthesis of an n-qubit stabilizer state for linear-nearest neighbour connectivity, in 2-qubit depth 2*n+2 and two distinct CX layers, using CX and phase gates (S, Sdg or Z).

Parameters

stab (StabilizerState) – a stabilizer state.

Returns

a circuit implementation of the stabilizer state.

Return type

QuantumCircuit

Reference:

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph] (opens in a new tab)
2. Dmitri Maslov, Martin Roetteler, Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations, arXiv:1705.09176 (opens in a new tab).

Discrete Basis Synthesis

qiskit.synthesis.generate_basic_approximations(basis_gates, depth, filename=None)

Generates a list of GateSequence``s with the gates in ``basic_gates.

Parameters

• basis_gates (list (opens in a new tab)[str (opens in a new tab) |Gate]) – The gates from which to create the sequences of gates.
• depth (int (opens in a new tab)) – The maximum depth of the approximations.
• filename (str (opens in a new tab) | None) – If provided, the basic approximations are stored in this file.

Returns

List of GateSequences using the gates in basic_gates.

Raises

ValueError (opens in a new tab) – If basis_gates contains an invalid gate identifier.

Return type

list (opens in a new tab)[GateSequence]