---
title: synthesis (v1.1)
description: API reference for qiskit.synthesis in qiskit v1.1
source: https://eu-de.quantum.cloud.ibm.com/docs/en/api/qiskit/1.1/synthesis
---

# Circuit Synthesis

`qiskit.synthesis`

## Evolution Synthesis

|                                                                                                                                 |                                                                                                                                               |
| ------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------- |
| [`EvolutionSynthesis`](/docs/api/qiskit/1.1/qiskit.synthesis.EvolutionSynthesis "qiskit.synthesis.EvolutionSynthesis")()        | Interface for evolution synthesis algorithms.                                                                                                 |
| [`ProductFormula`](/docs/api/qiskit/1.1/qiskit.synthesis.ProductFormula "qiskit.synthesis.ProductFormula")(order\[, reps, ...]) | Product formula base class for the decomposition of non-commuting operator exponentials.                                                      |
| [`LieTrotter`](/docs/api/qiskit/1.1/qiskit.synthesis.LieTrotter "qiskit.synthesis.LieTrotter")(\[reps, insert\_barriers, ...])  | The Lie-Trotter product formula.                                                                                                              |
| [`SuzukiTrotter`](/docs/api/qiskit/1.1/qiskit.synthesis.SuzukiTrotter "qiskit.synthesis.SuzukiTrotter")(\[order, reps, ...])    | The (higher order) Suzuki-Trotter product formula.                                                                                            |
| [`MatrixExponential`](/docs/api/qiskit/1.1/qiskit.synthesis.MatrixExponential "qiskit.synthesis.MatrixExponential")()           | Exact operator evolution via matrix exponentiation and unitary synthesis.                                                                     |
| [`QDrift`](/docs/api/qiskit/1.1/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

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/linear/cnot_synth.py#L26-L72)

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*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*bool*](https://docs.python.org/3/library/functions.html#bool)*]] | np.ndarray\[*[*bool*](https://docs.python.org/3/library/functions.html#bool)*]*) – $n \times n$ boolean invertible matrix, describing the state of the input circuit
- **section\_size** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The size of each section in the Patel–Markov–Hayes algorithm \[1].

**Returns**

a CX-only circuit implementing the linear transformation.

**Return type**

[QuantumCircuit](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")

**Raises**

[**QiskitError**](/docs/api/qiskit/1.1/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)

### synth\_cnot\_depth\_line\_kms

`qiskit.synthesis.synth_cnot_depth_line_kms(mat)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/linear/linear_depth_lnn.py#L241-L276)

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*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray)*\[*[*bool*](https://docs.python.org/3/library/functions.html#bool)*]*) – A boolean invertible matrix.

**Returns**

The synthesized quantum circuit.

**Raises**

[**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if `mat` is not invertible.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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](https://arxiv.org/abs/quant-ph/0701194)

## Linear-Phase Synthesis

### synth\_cz\_depth\_line\_mr

`qiskit.synthesis.synth_cz_depth_line_mr(mat)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/linear_phase/cz_depth_lnn.py#L121-L194)

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 [`CXGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate")s and phase gates ([`SGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.SGate "qiskit.circuit.library.SGate"), [`SdgGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.SdgGate "qiskit.circuit.library.SdgGate") or [`ZGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.ZGate "qiskit.circuit.library.ZGate")).

**Parameters**

**mat** ([*ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.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*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")

**References**

1. Dmitri Maslov, Martin Roetteler, *Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations*, [arXiv:1705.09176](https://arxiv.org/abs/1705.09176).

### synth\_cx\_cz\_depth\_line\_my

`qiskit.synthesis.synth_cx_cz_depth_line_my(mat_x, mat_z)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/linear_phase/cx_cz_depth_lnn.py#L218-L262)

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)) – a boolean symmetric matrix representing a CZ circuit. `mat_z[i][j]=1` represents a `cz(i,j)` gate
- **mat\_x** ([*ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.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*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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](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).

### synth\_cnot\_phase\_aam

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/linear_phase/cnot_phase_synth.py#L27-L188)

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*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*]]*) –

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

  ```python
  [[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:

  ```python
  x1^x2 + x0 + x0^x3 + x0^x1^x2 + x0^x1^x3 + x0^x1
  ```

- **angles** ([*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*str*](https://docs.python.org/3/library/stdtypes.html#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*](https://docs.python.org/3/library/functions.html#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**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – when dimensions of `cnots` and `angles` don’t align.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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](https://arxiv.org/abs/1712.01859)

## Permutation Synthesis

### synth\_permutation\_depth\_lnn\_kms

`qiskit.synthesis.synth_permutation_depth_lnn_kms(pattern)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/permutation/permutation_lnn.py#L21-L61)

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*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*] | np.ndarray\[*[*int*](https://docs.python.org/3/library/functions.html#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](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")

**References**

1. Samuel A. Kutin, David Petrie Moulton and Lawren M. Smithline. *Computation at a distance.*, [arXiv:quant-ph/0701194v1](https://arxiv.org/abs/quant-ph/0701194)

### synth\_permutation\_basic

`qiskit.synthesis.synth_permutation_basic(pattern)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/permutation/permutation_full.py#L27-L57)

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*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*] | np.ndarray\[*[*int*](https://docs.python.org/3/library/functions.html#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](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")

### synth\_permutation\_acg

`qiskit.synthesis.synth_permutation_acg(pattern)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/permutation/permutation_full.py#L60-L102)

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*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*] | np.ndarray\[*[*int*](https://docs.python.org/3/library/functions.html#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](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.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](https://doi.org/10.1145/167088.167239)
2. N. Alon, F. R. K. Chung, and R. L. Graham. *Routing Permutations on Graphs Via Matchings.*, [(Full paper)](https://www.cs.tau.ac.il/~nogaa/PDFS/r.pdf)

### synth\_permutation\_reverse\_lnn\_kms

`qiskit.synthesis.synth_permutation_reverse_lnn_kms(num_qubits)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/permutation/permutation_reverse_lnn.py#L66-L90)

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*](https://docs.python.org/3/library/functions.html#int)) – The number of qubits.

**Returns**

The synthesized quantum circuit.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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](https://arxiv.org/abs/quant-ph/0701194)

## Clifford Synthesis

### synth\_clifford\_full

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/clifford/clifford_decompose_full.py#L24-L64)

Decompose a [`Clifford`](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford") operator into a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.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*](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")) – A Clifford operator.
- **method** ([*str*](https://docs.python.org/3/library/stdtypes.html#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](/docs/api/qiskit/1.1/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)

### synth\_clifford\_ag

`qiskit.synthesis.synth_clifford_ag(clifford)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/clifford/clifford_decompose_ag.py#L36-L80)

Decompose a [`Clifford`](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford") operator into a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit") based on Aaronson-Gottesman method \[1].

**Parameters**

**clifford** ([*Clifford*](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.operators.symplectic.clifford.Clifford")) – A Clifford operator.

**Returns**

A circuit implementation of the Clifford.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")

**References**

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)

### synth\_clifford\_bm

`qiskit.synthesis.synth_clifford_bm(clifford)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/clifford/clifford_decompose_bm.py#L35-L85)

Optimal CX-cost decomposition of a [`Clifford`](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford") operator on 2 qubits or 3 qubits into a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit") based on the Bravyi-Maslov method \[1].

**Parameters**

**clifford** ([*Clifford*](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.operators.symplectic.clifford.Clifford")) – A Clifford operator.

**Returns**

A circuit implementation of the Clifford.

**Raises**

[**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if Clifford is on more than 3 qubits.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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)

### synth\_clifford\_greedy

`qiskit.synthesis.synth_clifford_greedy(clifford)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/clifford/clifford_decompose_greedy.py#L34-L124)

Decompose a [`Clifford`](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford") operator into a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.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*](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.operators.symplectic.clifford.Clifford")) – A Clifford operator.

**Returns**

A circuit implementation of the Clifford.

**Raises**

[**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if symplectic Gaussian elimination fails.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")

**References**

1. Sergey Bravyi, Shaohan Hu, Dmitri Maslov, Ruslan Shaydulin, *Clifford Circuit Optimization with Templates and Symbolic Pauli Gates*, [arXiv:2105.02291 \[quant-ph\]](https://arxiv.org/abs/2105.02291)

### synth\_clifford\_layers

`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)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/clifford/clifford_decompose_layers.py#L66-L167)

Synthesis of a [`Clifford`](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.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:

```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*](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")) – A Clifford operator.
- **cx\_synth\_func** (*Callable\[\[np.ndarray],* [*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")*]*) – A function to decompose the CX sub-circuit. It gets as input a boolean invertible matrix, and outputs a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit").
- **cz\_synth\_func** (*Callable\[\[np.ndarray],* [*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")*]*) – A function to decompose the CZ sub-circuit. It gets as input a boolean symmetric matrix, and outputs a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.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()`](#qiskit.synthesis.synth_cz_depth_line_mr "qiskit.synthesis.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](/docs/api/qiskit/1.1/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)

### synth\_clifford\_depth\_lnn

`qiskit.synthesis.synth_clifford_depth_lnn(cliff)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/clifford/clifford_decompose_layers.py#L414-L444)

Synthesis of a [`Clifford`](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford") into layers for linear-nearest neighbour 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*](/docs/api/qiskit/1.1/qiskit.quantum_info.Clifford "qiskit.quantum_info.Clifford")) – a Clifford operator.

**Returns**

a circuit implementation of the Clifford.

**Return type**

[QuantumCircuit](/docs/api/qiskit/1.1/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. 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).

## CNOTDihedral Synthesis

### synth\_cnotdihedral\_full

`qiskit.synthesis.synth_cnotdihedral_full(elem)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/cnotdihedral/cnotdihedral_decompose_full.py#L25-L52)

Decompose a [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element into a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.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*](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.operators.dihedral.dihedral.CNOTDihedral")) – A [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element.

**Returns**

A circuit implementation of the [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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).

### synth\_cnotdihedral\_two\_qubits

`qiskit.synthesis.synth_cnotdihedral_two_qubits(elem)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/cnotdihedral/cnotdihedral_decompose_two_qubits.py#L22-L266)

Decompose a [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element on a single qubit and two qubits into a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit"). This decomposition has an optimal number of [`CXGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate")s.

**Parameters**

**elem** ([*CNOTDihedral*](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.operators.dihedral.dihedral.CNOTDihedral")) – A [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element.

**Returns**

A circuit implementation of the [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element.

**Raises**

[**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if the element in not 1-qubit or 2-qubit [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral").

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")

**References**

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)

### synth\_cnotdihedral\_general

`qiskit.synthesis.synth_cnotdihedral_general(elem)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/cnotdihedral/cnotdihedral_decompose_general.py#L22-L141)

Decompose a [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element into a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit").

Decompose a general [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.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()`](#qiskit.synthesis.synth_cnotdihedral_two_qubits "qiskit.synthesis.synth_cnotdihedral_two_qubits").

**Parameters**

**elem** ([*CNOTDihedral*](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.operators.dihedral.dihedral.CNOTDihedral")) – A [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element.

**Returns**

A circuit implementation of the [`CNOTDihedral`](/docs/api/qiskit/1.1/qiskit.quantum_info.CNOTDihedral "qiskit.quantum_info.CNOTDihedral") element.

**Raises**

[**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if the element could not be decomposed into a circuit.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")

**References**

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

### synth\_stabilizer\_layers

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/stabilizer/stabilizer_decompose.py#L36-L119)

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:

```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*](/docs/api/qiskit/1.1/qiskit.quantum_info.StabilizerState "qiskit.quantum_info.states.stabilizerstate.StabilizerState")) – A stabilizer state.
- **cz\_synth\_func** ([*Callable*](https://docs.python.org/3/library/collections.abc.html#collections.abc.Callable)*\[\[*[*ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray)*],* [*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")*]*) – A function to decompose the CZ sub-circuit. It gets as input a boolean symmetric matrix, and outputs a [`QuantumCircuit`](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit").
- **cz\_func\_reverse\_qubits** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – `True` only if `cz_synth_func` is [`synth_cz_depth_line_mr()`](#qiskit.synthesis.synth_cz_depth_line_mr "qiskit.synthesis.synth_cz_depth_line_mr"), since this function returns a circuit that reverts the order of qubits.
- **validate** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – If `True`, validates the synthesis process.

**Returns**

A circuit implementation of the stabilizer state.

**Raises**

[**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.QiskitError") – if the input is not a [`StabilizerState`](/docs/api/qiskit/1.1/qiskit.quantum_info.StabilizerState "qiskit.quantum_info.StabilizerState").

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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)

### synth\_stabilizer\_depth\_lnn

`qiskit.synthesis.synth_stabilizer_depth_lnn(stab)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/stabilizer/stabilizer_decompose.py#L168-L193)

Synthesis of an n-qubit stabilizer state for linear-nearest neighbour connectivity, in 2-qubit depth $2n+2$ and two distinct CX layers, using [`CXGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate")s and phase gates ([`SGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.SGate "qiskit.circuit.library.SGate"), [`SdgGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.SdgGate "qiskit.circuit.library.SdgGate") or [`ZGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.ZGate "qiskit.circuit.library.ZGate")).

**Parameters**

**stab** ([*StabilizerState*](/docs/api/qiskit/1.1/qiskit.quantum_info.StabilizerState "qiskit.quantum_info.states.stabilizerstate.StabilizerState")) – A stabilizer state.

**Returns**

A circuit implementation of the stabilizer state.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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. Dmitri Maslov, Martin Roetteler, *Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations*, [arXiv:1705.09176](https://arxiv.org/abs/1705.09176).

### synth\_circuit\_from\_stabilizers

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/stabilizer/stabilizer_circuit.py#L27-L149)

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*](https://docs.python.org/3/library/collections.abc.html#collections.abc.Collection)*\[*[*str*](https://docs.python.org/3/library/stdtypes.html#str)*]*) – List of stabilizer strings
- **allow\_redundant** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – Allow redundant stabilizers (i.e., some stabilizers can be products of the others)
- **allow\_underconstrained** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – Allow underconstrained set of stabilizers (i.e., the stabilizers do not specify a unique state)
- **invert** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – Return inverse circuit

**Returns**

A circuit that generates a state stabilized by `stabilizers`.

**Raises**

[**QiskitError**](/docs/api/qiskit/1.1/exceptions#qiskit.exceptions.QiskitError "qiskit.exceptions.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*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")

**References**

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

## Discrete Basis Synthesis

|                                                                                                                                                        |                                                      |
| ------------------------------------------------------------------------------------------------------------------------------------------------------ | ---------------------------------------------------- |
| [`SolovayKitaevDecomposition`](/docs/api/qiskit/1.1/qiskit.synthesis.SolovayKitaevDecomposition "qiskit.synthesis.SolovayKitaevDecomposition")(\[...]) | The Solovay Kitaev discrete decomposition algorithm. |

### generate\_basic\_approximations

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/discrete_basis/generate_basis_approximations.py#L121-L163)

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

**Parameters**

- **basis\_gates** ([*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*str*](https://docs.python.org/3/library/stdtypes.html#str)  *|*[*Gate*](/docs/api/qiskit/1.1/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)) – The maximum depth of the approximations.
- **filename** ([*str*](https://docs.python.org/3/library/stdtypes.html#str) *| None*) – If provided, the basic approximations are stored in this file.

**Returns**

List of `GateSequence`s using the gates in `basis_gates`.

**Raises**

[**ValueError**](https://docs.python.org/3/library/exceptions.html#ValueError) – If `basis_gates` contains an invalid gate identifier.

**Return type**

[list](https://docs.python.org/3/library/stdtypes.html#list)\[GateSequence]

## Basis Change Synthesis

### synth\_qft\_line

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/qft/qft_decompose_lnn.py#L21-L70)

Synthesis of a QFT circuit for a linear nearest neighbor connectivity. Based on Fig 2.b in Fowler et al. \[1].

Note that this method *reverts* the order of qubits in the circuit, compared to the original [`QFT`](/docs/api/qiskit/1.1/qiskit.circuit.library.QFT "qiskit.circuit.library.QFT") code. Hence, the default value of the `do_swaps` parameter is `True` since it produces a circuit with fewer CX gates.

**Parameters**

- **num\_qubits** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The number of qubits on which the QFT acts.
- **approximation\_degree** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The degree of approximation (0 for no approximation).
- **do\_swaps** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – Whether to include the final swaps in the QFT.

**Returns**

A circuit implementation of the QFT circuit.

**Return type**

[*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.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 \[quant-ph\]](https://arxiv.org/abs/quant-ph/0402196)

## Unitary Synthesis

Decomposition of general $2^n \times 2^n$ unitary matrices for any number of qubits.

### qs\_decomposition

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.1/qiskit/synthesis/unitary/qsd.py#L35-L148)

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

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

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

The number of [`CXGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate")s generated with the decomposition without optimizations is:

$$
\frac{9}{16} 4^n - \frac{3}{2} 2^n
$$

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

$$
\frac{1}{3} 4^{n - 2} - 1.
$$

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

$$
4^{n-2} - 1.
$$

**Parameters**

- **mat** (*np.ndarray*) – unitary matrix to decompose
- **opt\_a1** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – whether to try optimization A.1 from Shende et al. \[1]. This should eliminate 1 `cx` per call. If `True`, [`CZGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CZGate "qiskit.circuit.library.CZGate")s are left in the output. If desired these can be further decomposed to [`CXGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate")s.
- **opt\_a2** ([*bool*](https://docs.python.org/3/library/functions.html#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*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")*] | None*) – optional 1Q decomposer. If None, uses [`OneQubitEulerDecomposer`](/docs/api/qiskit/1.1/qiskit.synthesis.OneQubitEulerDecomposer "qiskit.synthesis.OneQubitEulerDecomposer").
- **decomposer\_2q** (*Callable\[\[np.ndarray],* [*QuantumCircuit*](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")*] | None*) – optional 2Q decomposer. If None, uses [`TwoQubitBasisDecomposer`](/docs/api/qiskit/1.1/qiskit.synthesis.TwoQubitBasisDecomposer "qiskit.synthesis.TwoQubitBasisDecomposer").

**Returns**

Decomposed quantum circuit.

**Return type**

[QuantumCircuit](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")

**References**

1. Shende, Bullock, Markov, *Synthesis of Quantum Logic Circuits*, [arXiv:0406176 \[quant-ph\]](https://arxiv.org/abs/quant-ph/0406176)

The Approximate Quantum Compiler is available as the module [`qiskit.synthesis.unitary.aqc`](/docs/api/qiskit/1.1/qiskit.synthesis.unitary.aqc#module-qiskit.synthesis.unitary.aqc "qiskit.synthesis.unitary.aqc").

## One-Qubit Synthesis

|                                                                                                                                                           |                                                                       |
| --------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------- |
| [`OneQubitEulerDecomposer`](/docs/api/qiskit/1.1/qiskit.synthesis.OneQubitEulerDecomposer "qiskit.synthesis.OneQubitEulerDecomposer")(\[basis, use\_dag]) | A class for decomposing 1-qubit unitaries into Euler angle rotations. |

## Two-Qubit Synthesis

|                                                                                                                                                                      |                                                                                                                                                                                     |
| -------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| [`TwoQubitBasisDecomposer`](/docs/api/qiskit/1.1/qiskit.synthesis.TwoQubitBasisDecomposer "qiskit.synthesis.TwoQubitBasisDecomposer")(gate\[, ...])                  | A class for decomposing 2-qubit unitaries into minimal number of uses of a 2-qubit basis gate.                                                                                      |
| [`XXDecomposer`](/docs/api/qiskit/1.1/qiskit.synthesis.XXDecomposer "qiskit.synthesis.XXDecomposer")(\[basis\_fidelity, euler\_basis, ...])                          | A class for optimal decomposition of 2-qubit unitaries into 2-qubit basis gates of `XX` type (i.e., each locally equivalent to $CAN(\alpha, 0, 0)$ for a possibly varying $alpha$). |
| [`TwoQubitWeylDecomposition`](/docs/api/qiskit/1.1/qiskit.synthesis.TwoQubitWeylDecomposition "qiskit.synthesis.TwoQubitWeylDecomposition")(unitary\_matrix\[, ...]) | Two-qubit Weyl decomposition.                                                                                                                                                       |

### two\_qubit\_cnot\_decompose

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

This is an instance of [`TwoQubitBasisDecomposer`](/docs/api/qiskit/1.1/qiskit.synthesis.TwoQubitBasisDecomposer "qiskit.synthesis.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*](/docs/api/qiskit/1.1/qiskit.quantum_info.Operator "qiskit.quantum_info.Operator") *or np.ndarray*) – The 4x4 unitary to synthesize.
- **basis\_fidelity** ([*float*](https://docs.python.org/3/library/functions.html#float) *or None*) – If given the assumed fidelity for applications of [`CXGate`](/docs/api/qiskit/1.1/qiskit.circuit.library.CXGate "qiskit.circuit.library.CXGate").
- **approximate** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – If `True` approximate if `basis_fidelity` is less than 1.0.

**Returns**

The synthesized circuit of the input unitary.

**Return type**

[QuantumCircuit](/docs/api/qiskit/1.1/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")