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---
title: synthesis
description: API reference for qiskit.synthesis
in_page_toc_min_heading_level: 2
python_api_type: module
python_api_name: qiskit.synthesis
---
<span id="module-qiskit.synthesis" />
<span id="qiskit-synthesis" />
<span id="circuit-synthesis-qiskit-synthesis" />
# Circuit Synthesis
<span id="module-qiskit.synthesis" />
`qiskit.synthesis`
<span id="id1" />
## Evolution Synthesis
| | |
| ---------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------- |
| [`EvolutionSynthesis`](qiskit.synthesis.EvolutionSynthesis "qiskit.synthesis.EvolutionSynthesis")() | Interface for evolution synthesis algorithms. |
| [`ProductFormula`](qiskit.synthesis.ProductFormula "qiskit.synthesis.ProductFormula")(order\[, reps, ...]) | Product formula base class for the decomposition of non-commuting operator exponentials. |
| [`LieTrotter`](qiskit.synthesis.LieTrotter "qiskit.synthesis.LieTrotter")(\[reps, insert\_barriers, ...]) | The Lie-Trotter product formula. |
| [`SuzukiTrotter`](qiskit.synthesis.SuzukiTrotter "qiskit.synthesis.SuzukiTrotter")(\[order, reps, ...]) | The (higher order) Suzuki-Trotter product formula. |
| [`MatrixExponential`](qiskit.synthesis.MatrixExponential "qiskit.synthesis.MatrixExponential")() | Exact operator evolution via matrix exponentiation and unitary synthesis. |
| [`QDrift`](qiskit.synthesis.QDrift "qiskit.synthesis.QDrift")(\[reps, insert\_barriers, ...]) | The QDrift Trotterization method, which selects each each term in the Trotterization randomly, with a probability proportional to its weight. |
## Linear Function Synthesis
### synth\_cnot\_count\_full\_pmh
<Function id="qiskit.synthesis.synth_cnot_count_full_pmh" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/linear/cnot_synth.py" signature="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*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)")*\[*[*list*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)")*] or ndarray*) – n x n boolean invertible matrix, describing the state of the input circuit
* **section\_size** ([*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.12)")) – 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](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Raises**
[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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 \[quant-ph\]](https://arxiv.org/abs/quant-ph/0302002)
</Function>
### synth\_cnot\_depth\_line\_kms
<Function id="qiskit.synthesis.synth_cnot_depth_line_kms" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/linear/linear_depth_lnn.py" signature="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](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Raises**
[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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](https://arxiv.org/abs/quant-ph/0701194)
</Function>
## Linear-Phase Synthesis
### synth\_cz\_depth\_line\_mr
<Function id="qiskit.synthesis.synth_cz_depth_line_mr" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/linear_phase/cz_depth_lnn.py" signature="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*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray "(in NumPy v1.26)")) – 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](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Reference:**
1. Dmitri Maslov, Martin Roetteler, *Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations*, [arXiv:1705.09176](https://arxiv.org/abs/1705.09176).
</Function>
### synth\_cx\_cz\_depth\_line\_my
<Function id="qiskit.synthesis.synth_cx_cz_depth_line_my" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/linear_phase/cx_cz_depth_lnn.py" signature="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*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray "(in NumPy v1.26)")) – a boolean symmetric matrix representing a CZ circuit. Mz\[i]\[j]=1 represents a CZ(i,j) gate
* **mat\_x** ([*ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray "(in NumPy v1.26)")) – 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](qiskit.circuit.QuantumCircuit "qiskit.circuit.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](https://arxiv.org/abs/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](https://arxiv.org/abs/2210.16195).
</Function>
## Permutation Synthesis
### synth\_permutation\_depth\_lnn\_kms
<Function id="qiskit.synthesis.synth_permutation_depth_lnn_kms" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/permutation/permutation_lnn.py" signature="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](https://arxiv.org/abs/quant-ph/0701194), 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*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)")*\[*[*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.12)")*], 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.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
</Function>
### synth\_permutation\_basic
<Function id="qiskit.synthesis.synth_permutation_basic" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/permutation/permutation_full.py" signature="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*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)")*\[*[*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.12)")*], 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.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
</Function>
### synth\_permutation\_acg
<Function id="qiskit.synthesis.synth_permutation_acg" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/permutation/permutation_full.py" signature="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](https://www.cs.tau.ac.il/~nogaa/PDFS/r.pdf).
**Parameters**
**pattern** (*Union\[*[*list*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)")*\[*[*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.12)")*], 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.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
</Function>
## Clifford Synthesis
### synth\_clifford\_full
<Function id="qiskit.synthesis.synth_clifford_full" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/clifford/clifford_decompose_full.py" signature="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*](qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")) – a clifford operator.
* **method** ([*str*](https://docs.python.org/3/library/stdtypes.html#str "(in Python v3.12)")) – 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](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**References**
1. S. Bravyi, D. Maslov, *Hadamard-free circuits expose the structure of the Clifford group*, [arXiv:2003.09412 \[quant-ph\]](https://arxiv.org/abs/2003.09412)
2. S. Aaronson, D. Gottesman, *Improved Simulation of Stabilizer Circuits*, Phys. Rev. A 70, 052328 (2004). [arXiv:quant-ph/0406196](https://arxiv.org/abs/quant-ph/0406196)
3. Sergey Bravyi, Shaohan Hu, Dmitri Maslov, Ruslan Shaydulin, *Clifford Circuit Optimization with Templates and Symbolic Pauli Gates*, [arXiv:2105.02291 \[quant-ph\]](https://arxiv.org/abs/2105.02291)
</Function>
### synth\_clifford\_ag
<Function id="qiskit.synthesis.synth_clifford_ag" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/clifford/clifford_decompose_ag.py" signature="qiskit.synthesis.synth_clifford_ag(clifford)">
Decompose a Clifford operator into a QuantumCircuit based on Aaronson-Gottesman method.
**Parameters**
**clifford** ([*Clifford*](qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")) – a clifford operator.
**Returns**
a circuit implementation of the Clifford.
**Return type**
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Reference:**
1. S. Aaronson, D. Gottesman, *Improved Simulation of Stabilizer Circuits*, Phys. Rev. A 70, 052328 (2004). [arXiv:quant-ph/0406196](https://arxiv.org/abs/quant-ph/0406196)
</Function>
### synth\_clifford\_bm
<Function id="qiskit.synthesis.synth_clifford_bm" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/clifford/clifford_decompose_bm.py" signature="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*](qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")) – a clifford operator.
**Returns**
a circuit implementation of the Clifford.
**Return type**
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Raises**
[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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 \[quant-ph\]](https://arxiv.org/abs/2003.09412)
</Function>
### synth\_clifford\_greedy
<Function id="qiskit.synthesis.synth_clifford_greedy" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/clifford/clifford_decompose_greedy.py" signature="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*](qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")) – a clifford operator.
**Returns**
a circuit implementation of the Clifford.
**Return type**
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Raises**
[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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 \[quant-ph\]](https://arxiv.org/abs/2105.02291)
</Function>
### synth\_clifford\_layers
<Function id="qiskit.synthesis.synth_clifford_layers" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/clifford/clifford_decompose_layers.py" signature="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:
```python
┌─────┐┌─────┐┌────────┐┌─────┐┌─────┐┌─────┐┌─────┐┌────────┐
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*](qiskit.quantum_info.Clifford "qiskit.quantum_info.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](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Reference:**
1. S. Bravyi, D. Maslov, *Hadamard-free circuits expose the structure of the Clifford group*, [arXiv:2003.09412 \[quant-ph\]](https://arxiv.org/abs/2003.09412)
</Function>
### synth\_clifford\_depth\_lnn
<Function id="qiskit.synthesis.synth_clifford_depth_lnn" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/clifford/clifford_decompose_layers.py" signature="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*](qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")) – a clifford operator.
**Returns**
a circuit implementation of the Clifford.
**Return type**
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Reference:**
1. S. Bravyi, D. Maslov, *Hadamard-free circuits expose the structure of the Clifford group*, [arXiv:2003.09412 \[quant-ph\]](https://arxiv.org/abs/2003.09412)
2. Dmitri Maslov, Martin Roetteler, *Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations*, [arXiv:1705.09176](https://arxiv.org/abs/1705.09176).
3. Dmitri Maslov, Willers Yang, *CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate*, [arXiv:2210.16195](https://arxiv.org/abs/2210.16195).
</Function>
## CNOTDihedral Synthesis
### synth\_cnotdihedral\_full
<Function id="qiskit.synthesis.synth_cnotdihedral_full" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/cnotdihedral/cnotdihedral_decompose_full.py" signature="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*](qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")) – a CNOTDihedral element.
**Returns**
a circuit implementation of the CNOTDihedral element.
**Return type**
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.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](https://quantum-journal.org/papers/q-2020-12-07-369/)
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).
</Function>
### synth\_cnotdihedral\_two\_qubits
<Function id="qiskit.synthesis.synth_cnotdihedral_two_qubits" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/cnotdihedral/cnotdihedral_decompose_two_qubits.py" signature="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*](qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")) – a CNOTDihedral element.
**Returns**
a circuit implementation of the CNOTDihedral element.
**Return type**
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Raises**
[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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 \[quant-ph\]](https://arxiv.org/abs/2006.12042)
</Function>
### synth\_cnotdihedral\_general
<Function id="qiskit.synthesis.synth_cnotdihedral_general" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/cnotdihedral/cnotdihedral_decompose_general.py" signature="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*](qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral")) – a CNOTDihedral element.
**Returns**
a circuit implementation of the CNOTDihedral element.
**Return type**
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Raises**
[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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).
</Function>
## Stabilizer State Synthesis
### synth\_stabilizer\_layers
<Function id="qiskit.synthesis.synth_stabilizer_layers" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/stabilizer/stabilizer_decompose.py" signature="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:
```python
┌─────┐┌─────┐┌─────┐┌─────┐┌────────┐
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*](qiskit.quantum_info.StabilizerState "qiskit.quantum_info.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](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Raises**
[**QiskitError**](exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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 \[quant-ph\]](https://arxiv.org/abs/2003.09412)
</Function>
### synth\_stabilizer\_depth\_lnn
<Function id="qiskit.synthesis.synth_stabilizer_depth_lnn" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/stabilizer/stabilizer_decompose.py" signature="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*](qiskit.quantum_info.StabilizerState "qiskit.quantum_info.StabilizerState")) – a stabilizer state.
**Returns**
a circuit implementation of the stabilizer state.
**Return type**
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
**Reference:**
1. S. Bravyi, D. Maslov, *Hadamard-free circuits expose the structure of the Clifford group*, [arXiv:2003.09412 \[quant-ph\]](https://arxiv.org/abs/2003.09412)
2. Dmitri Maslov, Martin Roetteler, *Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations*, [arXiv:1705.09176](https://arxiv.org/abs/1705.09176).
</Function>
## Discrete Basis Synthesis
| | |
| --------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------- |
| [`SolovayKitaevDecomposition`](qiskit.synthesis.SolovayKitaevDecomposition "qiskit.synthesis.SolovayKitaevDecomposition")(\[...]) | The Solovay Kitaev discrete decomposition algorithm. |
### generate\_basic\_approximations
<Function id="qiskit.synthesis.generate_basic_approximations" github="https://github.com/qiskit/qiskit/tree/stable/0.25/qiskit/synthesis/discrete_basis/generate_basis_approximations.py" signature="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*](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)")*\[*[*str*](https://docs.python.org/3/library/stdtypes.html#str "(in Python v3.12)") *|*[*Gate*](qiskit.circuit.Gate "qiskit.circuit.Gate")*]*) – The gates from which to create the sequences of gates.
* **depth** ([*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.12)")) – The maximum depth of the approximations.
* **filename** ([*str*](https://docs.python.org/3/library/stdtypes.html#str "(in Python v3.12)") *| None*) – If provided, the basic approximations are stored in this file.
**Returns**
List of `GateSequences` using the gates in `basic_gates`.
**Raises**
[**ValueError**](https://docs.python.org/3/library/exceptions.html#ValueError "(in Python v3.12)") – If `basis_gates` contains an invalid gate identifier.
**Return type**
[list](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.12)")\[GateSequence]
</Function>
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