Circuit Synthesis

qiskit.synthesis.synth_cnot_count_full_pmh(state, section_size=None)

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 $n \times n$ matrix.

Parameters

• state (list[list[bool]] | np.ndarray[bool]) – $n \times n$ boolean invertible matrix, describing the state of the input circuit.
• section_size (int | None) – The size of each section in the Patel–Markov–Hayes algorithm [1]. If None it is chosen to be $\max(2, \alpha\log_2(n))$ with $\alpha = 0.56$, which approximately minimizes the upper bound on the number of row operations given in [1] Eq. (3).

Returns

A CX-only circuit implementing the linear transformation.

Raises

ValueError – When section_size is larger than the number of columns.

Return type

QuantumCircuit

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]

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], section 7. This algorithm synthesizes any linear reversible circuit of $n$ qubits over a linear nearest-neighbor architecture using CX gates with depth at most $5n$.

Parameters

mat (ndarray[bool]) – A boolean invertible matrix.

Returns

The synthesized quantum circuit.

Raises

QiskitError – if mat is not invertible.

Return type

QuantumCircuit

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

qiskit.synthesis.synth_cz_depth_line_mr(mat)

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

Note that this method reverts the order of qubits in the circuit, and returns a circuit containing CXGates and phase gates (SGate, SdgGate or ZGate).

Parameters

mat (ndarray) – 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 $2n+2$ for LNN connectivity.

Return type

QuantumCircuit

References

1. Dmitri Maslov, Martin Roetteler, Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations, arXiv:1705.09176.

qiskit.synthesis.synth_cx_cz_depth_line_my(mat_x, mat_z)

Joint synthesis of a -CZ-CX- circuit for linear nearest neighbor (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) – a boolean symmetric matrix representing a CZ circuit. mat_z[i][j]=1 represents a cz(i,j) gate
• mat_x (ndarray) – 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

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
2. Dmitri Maslov, Willers Yang, CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate, arXiv:2210.16195.

qiskit.synthesis.synth_cnot_phase_aam(cnots, angles, section_size=2)

This function is an implementation of the GraySynth algorithm of Amy, Azimadeh and Mosca.

GraySynth is a heuristic algorithm from [1] for synthesizing small parity networks. It is inspired by Gray codes. Given a set of binary strings $S$ (called cnots bellow), the algorithm synthesizes a parity network for $S$ by repeatedly choosing an index $i$ to expand and then effectively recursing on the co-factors $S_0$ and $S_1$, consisting of the strings $y \in S$, with $y_i = 0$ or $1$ respectively. As a subset $S$ is recursively expanded, cx gates are applied so that a designated target bit contains the (partial) parity $\chi_y(x)$ where $y_i = 1$ if and only if $y'_i = 1$ for all $y' \in S$. If $S$ contains a single element $\{y'\}$, then $y = y'$, and the target bit contains the value $\chi_{y'}(x)$ as desired.

Notably, rather than uncomputing this sequence of cx (CNOT) gates when a subset $S$ is finished being synthesized, the algorithm maintains the invariant that the remaining parities to be computed are expressed over the current state of bits. This allows the algorithm to avoid the ‘backtracking’ inherent in uncomputing-based methods.

The algorithm is described in detail in section 4 of [1].

Parameters

• cnots (list[list[int]]) –

  A matrix whose columns are the parities to be synthesized e.g.:

  [[0, 1, 1, 1, 1, 1],
   [1, 0, 0, 1, 1, 1],
   [1, 0, 0, 1, 0, 0],
   [0, 0, 1, 0, 1, 0]]

  corresponds to:

  x1^x2 + x0 + x0^x3 + x0^x1^x2 + x0^x1^x3 + x0^x1

• angles (list[str]) – A list containing all the phase-shift gates which are to be applied, in the same order as in cnots. A number is interpreted as the angle of $p(angle)$, otherwise the elements have to be 't', 'tdg', 's', 'sdg' or 'z'.

• section_size (int) – The size of every section in the Patel–Markov–Hayes algorithm. section_size must be a factor of the number of qubits.

Returns

The decomposed quantum circuit.

Raises

QiskitError – when dimensions of cnots and angles don’t align.

Return type

QuantumCircuit

References

1. Matthew Amy, Parsiad Azimzadeh, and Michele Mosca. On the controlled-NOT complexity of controlled-NOT–phase circuits., Quantum Science and Technology 4.1 (2018): 015002. arXiv:1712.01859

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 [1], section 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 (list[int] | np.ndarray[int]) – 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

References

1. Samuel A. Kutin, David Petrie Moulton and Lawren M. Smithline. Computation at a distance., arXiv:quant-ph/0701194v1

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 (list[int] | np.ndarray[int]) – 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 Proposition 4.1 in reference [1] with the detailed proof given in Theorem 2 in reference [2]

Parameters

pattern (list[int] | np.ndarray[int]) – 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

References

1. N. Alon, F. R. K. Chung, and R. L. Graham. Routing Permutations on Graphs Via Matchings., Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing(1993). Pages 583–591. (Extended abstract) 10.1145/167088.167239
2. N. Alon, F. R. K. Chung, and R. L. Graham. Routing Permutations on Graphs Via Matchings., (Full paper)

qiskit.synthesis.synth_permutation_reverse_lnn_kms(num_qubits)

Synthesize reverse permutation for linear nearest-neighbor architectures using Kutin, Moulton, Smithline method.

Synthesis algorithm for reverse permutation from [1], section 5. This algorithm synthesizes the reverse permutation on $n$ qubits over a linear nearest-neighbor architecture using CX gates with depth $2 * n + 2$.

Parameters

num_qubits (int) – The number of qubits.

Returns

The synthesized quantum circuit.

Return type

QuantumCircuit

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

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

Decompose a Clifford operator into a QuantumCircuit.

For $N \leq 3$ 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 | None) – Optional, a synthesis method ('AG' or 'greedy'). If set this overrides optimal decomposition for $N \leq 3$ 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]
2. S. Aaronson, D. Gottesman, Improved Simulation of Stabilizer Circuits, Phys. Rev. A 70, 052328 (2004). arXiv:quant-ph/0406196
3. Sergey Bravyi, Shaohan Hu, Dmitri Maslov, Ruslan Shaydulin, Clifford Circuit Optimization with Templates and Symbolic Pauli Gates, arXiv:2105.02291 [quant-ph]

qiskit.synthesis.synth_clifford_ag(clifford)

Decompose a Clifford operator into a QuantumCircuit based on Aaronson-Gottesman method [1].

Parameters

clifford (Clifford) – A Clifford operator.

Returns

A circuit implementation of the Clifford.

Return type

QuantumCircuit

References

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

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 the Bravyi-Maslov method [1].

Parameters

clifford (Clifford) – A Clifford operator.

Returns

A circuit implementation of the Clifford.

Raises

QiskitError – if Clifford is on more than 3 qubits.

Return type

QuantumCircuit

References

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph]

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 [1].

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

Note that this function only implements the greedy Clifford compiler from Appendix A of [1], and not the templates and symbolic Pauli gates optimizations that are mentioned in the same paper.

Parameters

clifford (Clifford) – A Clifford operator.

Returns

A circuit implementation of the Clifford.

Raises

QiskitError – if symplectic Gaussian elimination fails.

Return type

QuantumCircuit

References

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

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 [1].

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[[np.ndarray], QuantumCircuit]) – A function to decompose the CX sub-circuit. It gets as input a boolean invertible matrix, and outputs a QuantumCircuit.
• cz_synth_func (Callable[[np.ndarray], QuantumCircuit]) – 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

References

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph]

qiskit.synthesis.synth_clifford_depth_lnn(cliff)

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

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

Parameters

cliff (Clifford) – a Clifford operator.

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]
2. Dmitri Maslov, Martin Roetteler, Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations, arXiv:1705.09176.
3. Dmitri Maslov, Willers Yang, CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate, arXiv:2210.16195.

qiskit.synthesis.synth_cnotdihedral_full(elem)

Decompose a CNOTDihedral element into a QuantumCircuit.

For $N \leq 2$ 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
2. Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin and Jay M. Gambetta, Scalable randomized 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 CXGates.

Parameters

elem (CNOTDihedral) – A CNOTDihedral element.

Returns

A circuit implementation of the CNOTDihedral element.

Raises

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

Return type

QuantumCircuit

References

1. Shelly Garion and Andrew W. Cross, On the structure of the CNOT-Dihedral group, arXiv:2006.12042 [quant-ph]

qiskit.synthesis.synth_cnotdihedral_general(elem)

Decompose a CNOTDihedral element into a QuantumCircuit.

Decompose a general CNOTDihedral elements. The number of CX 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.

Raises

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

Return type

QuantumCircuit

References

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

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 reference [1], 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[[ndarray], QuantumCircuit]) – A function to decompose the CZ sub-circuit. It gets as input a boolean symmetric matrix, and outputs a QuantumCircuit.
• cz_func_reverse_qubits (bool) – True only if cz_synth_func is synth_cz_depth_line_mr(), since this function returns a circuit that reverts the order of qubits.
• validate (bool) – If True, validates the synthesis process.

Returns

A circuit implementation of the stabilizer state.

Raises

QiskitError – if the input is not a StabilizerState.

Return type

QuantumCircuit

References

1. S. Bravyi, D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv:2003.09412 [quant-ph]

qiskit.synthesis.synth_stabilizer_depth_lnn(stab)

Synthesis of an n-qubit stabilizer state for linear-nearest neighbor connectivity, in 2-qubit depth $2n+2$ and two distinct CX layers, using CXGates and phase gates (SGate, SdgGate or ZGate).

Parameters

stab (StabilizerState) – A stabilizer state.

Returns

A circuit implementation of the stabilizer state.

Return type

QuantumCircuit

References

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

qiskit.synthesis.synth_circuit_from_stabilizers(stabilizers, allow_redundant=False, allow_underconstrained=False, invert=False)

Synthesis of a circuit that generates a state stabilized by the stabilizers using Gaussian elimination with Clifford gates. If the stabilizers are underconstrained, and allow_underconstrained is True, the circuit will output one of the states stabilized by the stabilizers. Based on stim implementation.

Parameters

• stabilizers (Collection[str]) – List of stabilizer strings
• allow_redundant (bool) – Allow redundant stabilizers (i.e., some stabilizers can be products of the others)
• allow_underconstrained (bool) – Allow underconstrained set of stabilizers (i.e., the stabilizers do not specify a unique state)
• invert (bool) – Return inverse circuit

Returns

A circuit that generates a state stabilized by stabilizers.

Raises

QiskitError – if the stabilizers are invalid, do not commute, or contradict each other, if the list is underconstrained and allow_underconstrained is False, or if the list is redundant and allow_redundant is False.

Return type

QuantumCircuit

References

1. https://github.com/quantumlib/Stim/blob/c0dd0b1c8125b2096cd54b6f72884a459e47fe3e/src/stim/stabilizers/conversions.inl#L469
2. https://quantumcomputing.stackexchange.com/questions/12721/how-to-calculate-destabilizer-group-of-toric-and-other-codes

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

Generates a list of GateSequences with the gates in basis_gates.

Parameters

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

Returns

List of GateSequences using the gates in basis_gates.

Raises

ValueError – If basis_gates contains an invalid gate identifier.

Return type

list[GateSequence]

qiskit.synthesis.synth_qft_line(num_qubits, do_swaps=True, approximation_degree=0)

Construct a circuit for the Quantum Fourier Transform using linear neighbor connectivity.

The construction is based on Fig 2.b in Fowler et al. [1].

Parameters

• num_qubits (int) – The number of qubits on which the Quantum Fourier Transform acts.
• approximation_degree (int) – The degree of approximation (0 for no approximation). It is possible to implement the QFT approximately by ignoring controlled-phase rotations with the angle beneath a threshold. This is discussed in more detail in https://arxiv.org/abs/quant-ph/9601018 or https://arxiv.org/abs/quant-ph/0403071.
• do_swaps (bool) – Whether to synthesize the “QFT” or the “QFT-with-reversal” operation.

Returns

A circuit implementing the QFT operation.

Return type

QuantumCircuit

References

1. A. G. Fowler, S. J. Devitt, and L. C. L. Hollenberg, Implementation of Shor’s algorithm on a linear nearest neighbour qubit array, Quantum Info. Comput. 4, 4 (July 2004), 237–251. arXiv:quant-ph/0402196 [quant-ph]

qiskit.synthesis.synth_qft_full(num_qubits, do_swaps=True, approximation_degree=0, insert_barriers=False, inverse=False, name=None)

Construct a circuit for the Quantum Fourier Transform using all-to-all connectivity.

Parameters

• num_qubits (int) – The number of qubits on which the Quantum Fourier Transform acts.
• do_swaps (bool) – Whether to synthesize the “QFT” or the “QFT-with-reversal” operation.
• approximation_degree (int) – The degree of approximation (0 for no approximation). It is possible to implement the QFT approximately by ignoring controlled-phase rotations with the angle beneath a threshold. This is discussed in more detail in https://arxiv.org/abs/quant-ph/9601018 or https://arxiv.org/abs/quant-ph/0403071.
• insert_barriers (bool) – If True, barriers are inserted for improved visualization.
• inverse (bool) – If True, the inverse Quantum Fourier Transform is constructed.
• name (str | None) – The name of the circuit.

Returns

A circuit implementing the QFT operation.

Return type

QuantumCircuit

qiskit.synthesis.qs_decomposition(mat, opt_a1=True, opt_a2=True, decomposer_1q=None, decomposer_2q=None, *, _depth=0)

Decomposes a unitary matrix into one and two qubit gates using Quantum Shannon Decomposition,

This decomposition is described in Shende et al. [1].

  ┌───┐               ┌───┐     ┌───┐     ┌───┐
 ─┤   ├─       ───────┤ Rz├─────┤ Ry├─────┤ Rz├─────
  │   │    ≃     ┌───┐└─┬─┘┌───┐└─┬─┘┌───┐└─┬─┘┌───┐
/─┤   ├─       /─┤   ├──□──┤   ├──□──┤   ├──□──┤   ├
  └───┘          └───┘     └───┘     └───┘     └───┘

The number of CXGates generated with the decomposition without optimizations is:

If opt_a1 = True, the default, the CX count is reduced by:

If opt_a2 = True, the default, the CX count is reduced by:

Parameters

• mat (np.ndarray) – unitary matrix to decompose
• opt_a1 (bool) – whether to try optimization A.1 from Shende et al. [1]. This should eliminate 1 cx per call. If True, CZGates are left in the output. If desired these can be further decomposed to CXGates.
• opt_a2 (bool) – whether to try optimization A.2 from Shende et al. [1]. This decomposes two qubit unitaries into a diagonal gate and a two cx unitary and reduces overall cx count by $4^{n-2} - 1$.
• decomposer_1q (Callable[[np.ndarray], QuantumCircuit] | None) – optional 1Q decomposer. If None, uses OneQubitEulerDecomposer.
• decomposer_2q (Callable[[np.ndarray], QuantumCircuit] | None) – optional 2Q decomposer. If None, uses TwoQubitBasisDecomposer.

Returns

Decomposed quantum circuit.

Return type

QuantumCircuit

References

1. Shende, Bullock, Markov, Synthesis of Quantum Logic Circuits, arXiv:0406176 [quant-ph]

qiskit.synthesis.two_qubit_cnot_decompose(*args, **kwargs)

This is an instance of TwoQubitBasisDecomposer that always uses cx as the KAK gate for the basis decomposition. You can use this function as a quick access to cx-based 2-qubit decompositions.

Parameters

• unitary (Operator or np.ndarray) – The 4x4 unitary to synthesize.
• basis_fidelity (float or None) – If given the assumed fidelity for applications of CXGate.
• approximate (bool) – If True approximate if basis_fidelity is less than 1.0.

Returns

The synthesized circuit of the input unitary.

Return type

QuantumCircuit

qiskit.synthesis.synth_mcmt_vchain(gate, num_ctrl_qubits, num_target_qubits, ctrl_state=None)

Synthesize MCMT using a V-chain.

This uses a chain of CCX gates, using num_ctrl_qubits - 1 auxiliary qubits.

For example, a 3-control and 2-target H gate will be synthesized as:

q_0: ──■────────────────────────■──
       │                        │
q_1: ──■────────────────────────■──
       │                        │
q_2: ──┼────■──────────────■────┼──
       │    │  ┌───┐       │    │
q_3: ──┼────┼──┤ H ├───────┼────┼──
       │    │  └─┬─┘┌───┐  │    │
q_4: ──┼────┼────┼──┤ H ├──┼────┼──
     ┌─┴─┐  │    │  └─┬─┘  │  ┌─┴─┐
q_5: ┤ X ├──■────┼────┼────■──┤ X ├
     └───┘┌─┴─┐  │    │  ┌─┴─┐└───┘
q_6: ─────┤ X ├──■────■──┤ X ├─────
          └───┘          └───┘

Parameters

• gate (Gate) –
• num_ctrl_qubits (int) –
• num_target_qubits (int) –
• ctrl_state (int | None) –

Return type

QuantumCircuit

qiskit.synthesis.synth_mcx_n_dirty_i15(num_ctrl_qubits, relative_phase=False, action_only=False)

Synthesize a multi-controlled X gate with $k$ controls using $k - 2$ dirty ancillary qubits producing a circuit with $2 * k - 1$ qubits and at most $8 * k - 6$ CX gates, by Iten et. al. [1].

Parameters

• num_ctrl_qubits (int) – The number of control qubits.
• relative_phase (bool) – when set to True, the method applies the optimized multi-controlled X gate up to a relative phase, in a way that, by lemma 8 of [1], the relative phases of the action part cancel out with the phases of the reset part.
• action_only (bool) – when set to True, the method applies only the action part of lemma 8 of [1].

Returns

The synthesized quantum circuit.

Return type

QuantumCircuit

References

1. Iten et. al., Quantum Circuits for Isometries, Phys. Rev. A 93, 032318 (2016), arXiv:1501.06911

qiskit.synthesis.synth_mcx_n_clean_m15(num_ctrl_qubits)

Synthesize a multi-controlled X gate with $k$ controls using $k - 2$ clean ancillary qubits with producing a circuit with $2 * k - 1$ qubits and at most $6 * k - 6$ CX gates, by Maslov [1].

Parameters

num_ctrl_qubits (int) – The number of control qubits.

Returns

The synthesized quantum circuit.

Return type

QuantumCircuit

References

1. Maslov., Phys. Rev. A 93, 022311 (2016), arXiv:1508.03273

qiskit.synthesis.synth_mcx_1_clean_b95(num_ctrl_qubits)

Synthesize a multi-controlled X gate with $k$ controls using a single clean ancillary qubit producing a circuit with $k + 2$ qubits and at most $16 * k - 8$ CX gates, by Barenco et al. [1].

Parameters

num_ctrl_qubits (int) – The number of control qubits.

Returns

The synthesized quantum circuit.

Return type

QuantumCircuit

References

1. Barenco et. al., Phys.Rev. A52 3457 (1995), arXiv:quant-ph/9503016

qiskit.synthesis.synth_mcx_noaux_v24(num_ctrl_qubits)

Synthesize a multi-controlled X gate with $k$ controls based on the implementation for MCPhaseGate.

In turn, the MCPhase gate uses the decomposition for multi-controlled special unitaries described in [1].

Produces a quantum circuit with $k + 1$ qubits. The number of CX-gates is quadratic in $k$.

Parameters

num_ctrl_qubits (int) – The number of control qubits.

Returns

The synthesized quantum circuit.

Return type

QuantumCircuit

References

1. Vale et. al., Circuit Decomposition of Multicontrolled Special Unitary Single-Qubit Gates, IEEE TCAD 43(3) (2024), arXiv:2302.06377

qiskit.synthesis.synth_mcx_gray_code(num_ctrl_qubits)

Synthesize a multi-controlled X gate with $k$ controls using the Gray code.

Produces a quantum circuit with $k + 1$ qubits. This method produces exponentially many CX gates and should be used only for small values of $k$.

Parameters

num_ctrl_qubits (int) – The number of control qubits.

Returns

The synthesized quantum circuit.

Return type

QuantumCircuit

qiskit.synthesis.synth_c3x()

Efficient synthesis of 3-controlled X-gate.

Return type

QuantumCircuit

qiskit.synthesis.synth_c4x()

Efficient synthesis of 4-controlled X-gate.

Return type

QuantumCircuit

qiskit.synthesis.adder_qft_d00(num_state_qubits, kind='half')

A circuit that uses QFT to perform in-place addition on two qubit registers.

For registers with $n$ qubits, the QFT adder can perform addition modulo $2^n$ (with kind="fixed") or ordinary addition by adding a carry qubits (with kind="half"). The fixed adder uses $(3n^2 - n)/2$ CPhaseGate operators, with an additional $n$ for the half adder.

As an example, a non-fixed_point QFT adder circuit that performs addition on two 2-qubit sized registers is as follows:

 a_0: ─────────■──────■────────■──────────────────────────────────
               │      │        │
 a_1: ─────────┼──────┼────────┼────────■──────■──────────────────
      ┌──────┐ │      │        │P(π/4)  │      │P(π/2) ┌─────────┐
 b_0: ┤0     ├─┼──────┼────────■────────┼──────■───────┤0        ├
      │      │ │      │P(π/2)           │P(π)          │         │
 b_1: ┤1 Qft ├─┼──────■─────────────────■──────────────┤1 qft_dg ├
      │      │ │P(π)                                   │         │
cout: ┤2     ├─■───────────────────────────────────────┤2        ├
      └──────┘                                         └─────────┘

Parameters

• num_state_qubits (int) – The number of qubits in either input register for state $|a\rangle$ or $|b\rangle$. The two input registers must have the same number of qubits.
• kind (str) – The kind of adder, can be "half" for a half adder or "fixed" for a fixed-sized adder. A half adder contains a carry-out to represent the most-significant bit, but the fixed-sized adder doesn’t and hence performs addition modulo 2 ** num_state_qubits.

Return type

QuantumCircuit

References:

[1] T. G. Draper, Addition on a Quantum Computer, 2000. arXiv:quant-ph/0008033

[2] Ruiz-Perez et al., Quantum arithmetic with the Quantum Fourier Transform, 2017. arXiv:1411.5949

[3] Vedral et al., Quantum Networks for Elementary Arithmetic Operations, 1995. arXiv:quant-ph/9511018

qiskit.synthesis.adder_ripple_c04(num_state_qubits, kind='half')

A ripple-carry circuit to perform in-place addition on two qubit registers.

This circuit uses $2n + O(1)$ CCX gates and $5n + O(1)$ CX gates, at a depth of $2n + O(1)$ [1]. The constant depends on the kind of adder implemented.

As an example, a ripple-carry adder circuit that performs addition on two 3-qubit sized registers with a carry-in bit (kind="full") is as follows:

        ┌──────┐                                     ┌──────┐
 cin_0: ┤2     ├─────────────────────────────────────┤2     ├
        │      │┌──────┐                     ┌──────┐│      │
   a_0: ┤0     ├┤2     ├─────────────────────┤2     ├┤0     ├
        │      ││      │┌──────┐     ┌──────┐│      ││      │
   a_1: ┤  MAJ ├┤0     ├┤2     ├─────┤2     ├┤0     ├┤  UMA ├
        │      ││      ││      │     │      ││      ││      │
   a_2: ┤      ├┤  MAJ ├┤0     ├──■──┤0     ├┤  UMA ├┤      ├
        │      ││      ││      │  │  │      ││      ││      │
   b_0: ┤1     ├┤      ├┤  MAJ ├──┼──┤  UMA ├┤      ├┤1     ├
        └──────┘│      ││      │  │  │      ││      │└──────┘
   b_1: ────────┤1     ├┤      ├──┼──┤      ├┤1     ├────────
                └──────┘│      │  │  │      │└──────┘
   b_2: ────────────────┤1     ├──┼──┤1     ├────────────────
                        └──────┘┌─┴─┐└──────┘
cout_0: ────────────────────────┤ X ├────────────────────────
                                └───┘

Here MAJ and UMA gates correspond to the gates introduced in [1]. Note that in this implementation the input register qubits are ordered as all qubits from the first input register, followed by all qubits from the second input register.

Two different kinds of adders are supported. By setting the kind argument, you can also choose a half-adder, which doesn’t have a carry-in, and a fixed-sized-adder, which has neither carry-in nor carry-out, and thus acts on fixed register sizes. Unlike the full-adder, these circuits need one additional helper qubit.

The circuit diagram for the fixed-point adder (kind="fixed") on 3-qubit sized inputs is

        ┌──────┐┌──────┐                ┌──────┐┌──────┐
   a_0: ┤0     ├┤2     ├────────────────┤2     ├┤0     ├
        │      ││      │┌──────┐┌──────┐│      ││      │
   a_1: ┤      ├┤0     ├┤2     ├┤2     ├┤0     ├┤      ├
        │      ││      ││      ││      ││      ││      │
   a_2: ┤      ├┤  MAJ ├┤0     ├┤0     ├┤  UMA ├┤      ├
        │      ││      ││      ││      ││      ││      │
   b_0: ┤1 MAJ ├┤      ├┤  MAJ ├┤  UMA ├┤      ├┤1 UMA ├
        │      ││      ││      ││      ││      ││      │
   b_1: ┤      ├┤1     ├┤      ├┤      ├┤1     ├┤      ├
        │      │└──────┘│      ││      │└──────┘│      │
   b_2: ┤      ├────────┤1     ├┤1     ├────────┤      ├
        │      │        └──────┘└──────┘        │      │
help_0: ┤2     ├────────────────────────────────┤2     ├
        └──────┘                                └──────┘

It has one less qubit than the full-adder since it doesn’t have the carry-out, but uses a helper qubit instead of the carry-in, so it only has one less qubit, not two.

Parameters

• num_state_qubits (int) – The number of qubits in either input register for state $|a\rangle$ or $|b\rangle$. The two input registers must have the same number of qubits.
• kind (str) – The kind of adder, can be "full" for a full adder, "half" for a half adder, or "fixed" for a fixed-sized adder. A full adder includes both carry-in and carry-out, a half only carry-out, and a fixed-sized adder neither carry-in nor carry-out.

Raises

ValueError – If num_state_qubits is lower than 1.

Return type

QuantumCircuit

References:

[1] Cuccaro et al., A new quantum ripple-carry addition circuit, 2004. arXiv:quant-ph/0410184

[2] Vedral et al., Quantum Networks for Elementary Arithmetic Operations, 1995. arXiv:quant-ph/9511018

qiskit.synthesis.adder_ripple_v95(num_state_qubits, kind='half')

The VBE ripple carry adder [1].

This method uses $4n + O(1)$ CCX gates and $4n + 1$ CX gates at a depth of $6n - 2$ [2].

This circuit performs inplace addition of two equally-sized quantum registers. As an example, a classical adder circuit that performs full addition (i.e. including a carry-in bit) on two 2-qubit sized registers is as follows:

          ┌────────┐                       ┌───────────┐┌──────┐
   cin_0: ┤0       ├───────────────────────┤0          ├┤0     ├
          │        │                       │           ││      │
     a_0: ┤1       ├───────────────────────┤1          ├┤1     ├
          │        │┌────────┐     ┌──────┐│           ││  Sum │
     a_1: ┤        ├┤1       ├──■──┤1     ├┤           ├┤      ├
          │        ││        │  │  │      ││           ││      │
     b_0: ┤2 Carry ├┤        ├──┼──┤      ├┤2 Carry_dg ├┤2     ├
          │        ││        │┌─┴─┐│      ││           │└──────┘
     b_1: ┤        ├┤2 Carry ├┤ X ├┤2 Sum ├┤           ├────────
          │        ││        │└───┘│      ││           │
  cout_0: ┤        ├┤3       ├─────┤      ├┤           ├────────
          │        ││        │     │      ││           │
helper_0: ┤3       ├┤0       ├─────┤0     ├┤3          ├────────
          └────────┘└────────┘     └──────┘└───────────┘

Here Carry and Sum gates correspond to the gates introduced in [1]. Carry_dg correspond to the inverse of the Carry gate. Note that in this implementation the input register qubits are ordered as all qubits from the first input register, followed by all qubits from the second input register. This is different ordering as compared to Figure 2 in [1], which leads to a different drawing of the circuit.

Parameters

• num_state_qubits (int) – The size of the register.
• kind (str) – The kind of adder, can be "full" for a full adder, "half" for a half adder, or "fixed" for a fixed-sized adder. A full adder includes both carry-in and carry-out, a half only carry-out, and a fixed-sized adder neither carry-in nor carry-out.

Raises

ValueError – If num_state_qubits is lower than 1.

Return type

QuantumCircuit

References:

[1] Vedral et al., Quantum Networks for Elementary Arithmetic Operations, 1995. arXiv:quant-ph/9511018

[2] Cuccaro et al., A new quantum ripple-carry addition circuit, 2004. arXiv:quant-ph/0410184

qiskit.synthesis.multiplier_cumulative_h18(num_state_qubits, num_result_qubits=None)

A multiplication circuit to store product of two input registers out-of-place.

The circuit uses the approach from Ref. [1]. As an example, a multiplier circuit that performs a non-modular multiplication on two 3-qubit sized registers is:

from qiskit.synthesis.arithmetic import multiplier_cumulative_h18
 
num_state_qubits = 3
circuit = multiplier_cumulative_h18(num_state_qubits)
circuit.draw("mpl")

Multiplication in this circuit is implemented in a classical approach by performing a series of shifted additions using one of the input registers while the qubits from the other input register act as control qubits for the adders.

Parameters

• num_state_qubits (int) – The number of qubits in either input register for state $|a\rangle$ or $|b\rangle$. The two input registers must have the same number of qubits.
• num_result_qubits (int | None) – The number of result qubits to limit the output to. If number of result qubits is $n$, multiplication modulo $2^n$ is performed to limit the output to the specified number of qubits. Default value is 2 * num_state_qubits to represent any possible result from the multiplication of the two inputs.

Raises

ValueError – If num_result_qubits is given and not valid, meaning not in [num_state_qubits, 2 * num_state_qubits].

Return type

QuantumCircuit

References:

[1] Häner et al., Optimizing Quantum Circuits for Arithmetic, 2018. arXiv:1805.12445

qiskit.synthesis.multiplier_qft_r17(num_state_qubits, num_result_qubits=None)

A QFT multiplication circuit to store product of two input registers out-of-place.

Multiplication in this circuit is implemented using the procedure of Fig. 3 in [1], where weighted sum rotations are implemented as given in Fig. 5 in [1]. QFT is used on the output register and is followed by rotations controlled by input registers. The rotations transform the state into the product of two input registers in QFT base, which is reverted from QFT base using inverse QFT. For example, on 3 state qubits, a full multiplier is given by:

from qiskit.synthesis.arithmetic import multiplier_qft_r17
 
num_state_qubits = 3
circuit = multiplier_qft_r17(num_state_qubits)
circuit.draw("mpl")

Parameters

• num_state_qubits (int) – The number of qubits in either input register for state $|a\rangle$ or $|b\rangle$. The two input registers must have the same number of qubits.
• num_result_qubits (int | None) – The number of result qubits to limit the output to. If number of result qubits is $n$, multiplication modulo $2^n$ is performed to limit the output to the specified number of qubits. Default value is 2 * num_state_qubits to represent any possible result from the multiplication of the two inputs.

Raises

ValueError – If num_result_qubits is given and not valid, meaning not in [num_state_qubits, 2 * num_state_qubits].

Return type

QuantumCircuit

References:

[1] Ruiz-Perez et al., Quantum arithmetic with the Quantum Fourier Transform, 2017. arXiv:1411.5949

qiskit.synthesis.synth_weighted_sum_carry(weighted_sum)

Synthesize a weighted sum gate, by the number of state qubits and the qubit weights.

This method is described in Appendix A of [1].

Reference:

[1] Stamatopoulos et al. Option Pricing using Quantum Computers (2020)

Quantum 4, 291

Parameters

weighted_sum (WeightedSumGate) –

Return type

QuantumCircuit

qiskit.synthesis.synth_integer_comparator_2s(num_state_qubits, value, geq=True)

Implement an integer comparison based on 2s complement.

This is based on Appendix B of [1].

Parameters

• num_state_qubits (int) – The number of qubits encoding the value to compare to.
• value (int) – The value to compare to.
• geq (bool) – If True flip the target bit if the qubit state is $\geq$ than the value, otherwise implement $<$.

Returns

A circuit implementing the integer comparator.

Return type

QuantumCircuit

References

[1] J. Gacon et al. “Quantum-enhanced simulation-based optimization”

arXiv:2005.10780.

qiskit.synthesis.synth_integer_comparator_greedy(num_state_qubits, value, geq=True)

Implement an integer comparison based on value-by-value comparison.

For value smaller than 2 ** (num_state_qubits - 1) this circuit implements value multi-controlled gates with control states 0, 1, …, value - 1, such that the target qubit is flipped if the qubit state represents any of the allowed values. For value larger than that, geq is flipped. This implementation can require an exponential number of gates. If auxiliary qubits are available, the implementation provided by synth_integer_comparator_2s() is more efficient.

Parameters

• num_state_qubits (int) – The number of qubits encoding the value to compare to.
• value (int) – The value to compare to.
• geq (bool) – If True flip the target bit if the qubit state is $\geq$ than the value, otherwise implement $<$.

Returns

A circuit implementing the integer comparator.

Return type

QuantumCircuit