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This library contains special decompositions of particular gates into particular gate bases. It also contains functions for decomposing multiple controls.

For example, we provide particular decompositions of the Toffoli,
Fredkin, doubly-controlled *iX*-gate, and certain other controlled
Clifford gates, into the Clifford+*T* base. In some cases, we
provide more than one decomposition.

Many of these decompositions are taken or adapted from the literature, for example, from:

- M. A. Nielsen and I. L. Chuang,
*Quantum Computation and Quantum Information*, Cambridge University Press, 2002. - M. Amy, D. Maslov, M. Mosca, and M. Roetteler,
A meet-in-the-middle algorithm for fast synthesis
of depth-optimal quantum circuits,
*IEEE Transactions on Computer-Aided Design of**Integrated Circuits and Systems*32(6):818-830. Also available from http://arxiv.org/abs/1206.0758. - A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo,
N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter,
Elementary gates for quantum computation,
*Physical Review A*52(5):3457-3467, 1995. Also available from http://arxiv.org/abs/quantph/9503016. - P. Selinger, Quantum circuits of T-depth one,
*Physical Review A*87, 042302 (4 pages), 2013. Also available from http://arxiv.org/abs/1210.0974. - B. Giles and P. Selinger, Exact synthesis of multiqubit
Clifford+T circuits,
*Physical Review A*87, 032332 (7 pages), 2013. Also available from http://arxiv.org/abs/1212.0506.

- toffoli_NC_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- toffoli_AMMR_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- toffoli_V_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- toffoli_S_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- cc_iX_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- cc_iX_simple_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- cc_iX_S_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- ccZ_AMMR_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- ccZ_S_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- fredkin_at :: Qubit -> Qubit -> Signed Qubit -> Circ ()
- cH_AMMR_at :: Qubit -> Signed Qubit -> Circ ()
- controlled_W_at :: Qubit -> Qubit -> Signed Qubit -> Circ ()
- gate_W_CliffordT_at :: Qubit -> Qubit -> Circ ()
- controlled_iX_at :: Qubit -> Signed Qubit -> Circ ()
- controlled_S_at :: Qubit -> Signed Qubit -> Circ ()
- controlled_T_at :: Qubit -> Signed Qubit -> Circ ()
- controlled_V_at :: Qubit -> Signed Qubit -> Circ ()
- controlled_E_at :: Qubit -> Signed Qubit -> Circ ()
- controlled_YY_at :: Qubit -> Signed Qubit -> Circ ()
- toffoli_plain_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- cc_iX_plain_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ ()
- multi_cnot_barenco_at :: (Qubit -> Signed Qubit -> Signed Qubit -> Circ ()) -> (Qubit -> Signed Qubit -> Signed Qubit -> Circ ()) -> Qubit -> [Qubit] -> [Signed Qubit] -> Circ ()
- multi_ciX_noancilla_at :: (Qubit -> Signed Qubit -> Signed Qubit -> Circ ()) -> Qubit -> [Signed Qubit] -> Circ ()
- partition_controls :: [Signed Endpoint] -> ([Signed Qubit], [Signed Bit])
- with_signed_qubit :: Signed Qubit -> (Qubit -> Circ b) -> Circ b
- with_combined_controls :: (Qubit -> Signed Qubit -> Signed Qubit -> Circ ()) -> Int -> [Signed Endpoint] -> ([Signed Qubit] -> Circ a) -> Circ a

# Decomposition of gates

toffoli_NC_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the Toffoli gate into the Clifford+*T* base,
from Nielsen and Chuang (Figure 4.9). The first argument is the
target, and the other two are the controls. The controls can be
positive or negative.

toffoli_AMMR_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the Toffoli gate into the Clifford+*T* base,
from Amy et al. (http://arxiv.org/abs/1206.0758v3, Figure
13). The first argument is the target, and the other two are the
controls. The controls can be positive or negative.

toffoli_V_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the Toffoli gate using controlled Clifford operators, from Nielsen and Chuang (Figure 4.8). The controls can be positive or negative.

toffoli_S_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the Toffoli gate into the Clifford+*T* base,
using *T*-depth 1 and four ancillas. From
http://arxiv.org/abs/1210.0974 (Figure 1). The first argument is
the target, and the other two are the controls. The controls can be
positive or negative.

cc_iX_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the doubly-controlled *iX*-gate into the
Clifford+*T* base, using *T*-count 4 and *T*-depth 2. Adapted from
(http://arxiv.org/abs/1210.0974, Figure 10). The first argument
is the target, and the other two are the controls. The controls can
be positive or negative.

cc_iX_simple_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the doubly-controlled *iX*-gate into the
Clifford+*T* base, using *T*-count 4, and using the control qubits
only as controls. Derived from Nielsen and Chuang (Figure 4.9). The
controls can be positive or negative.

cc_iX_S_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the doubly-controlled *iX*-gate into the
Clifford+*T* base, using *T*-depth 1 and one ancilla. Adapted from
(http://arxiv.org/abs/1210.0974, Figure 9). The first argument
is the target, and the other two are the controls. The controls can
be positive or negative.

ccZ_AMMR_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the doubly-controlled *Z*-gate into the
Clifford+*T* base. Adapted from Amy et
al. (http://arxiv.org/abs/1206.0758v3, Figure 13). The controls can
be positive or negative.

ccZ_S_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the doubly-controlled *Z*-gate into the
Clifford+*T* base, using *T*-depth 1 and four ancillas. From
Selinger (http://arxiv.org/abs/1210.0974, Figure 1). The controls
can be positive or negative.

fredkin_at :: Qubit -> Qubit -> Signed Qubit -> Circ () Source #

Decomposition of the Fredkin (controlled-Swap) gate into the
Clifford+*T* base. The first two arguments are the targets, and the
last one the control. The controls can be positive or negative.

cH_AMMR_at :: Qubit -> Signed Qubit -> Circ () Source #

Decomposition of a controlled *H*-gate into the Clifford+*T*
base. From Amy et al. (http://arxiv.org/abs/1206.0758v3, Figure
5(a)). The first argument is the target and the second one the
control. The control can be positive or negative.

controlled_W_at :: Qubit -> Qubit -> Signed Qubit -> Circ () Source #

Decomposition of a controlled *W*-gate into the Clifford+*T*
base. The first two arguments are the targets, and the last
argument is the control. The control can be positive or negative.

gate_W_CliffordT_at :: Qubit -> Qubit -> Circ () Source #

Decomposition of a *W*-gate into the Clifford+*T* base.

controlled_iX_at :: Qubit -> Signed Qubit -> Circ () Source #

Decomposition of a controlled *iX*-gate into the Clifford+*T*
base. The first argument is the target, and the second one is the
control. The control can be positive or negative.

controlled_S_at :: Qubit -> Signed Qubit -> Circ () Source #

Decomposition of a controlled *S*-gate into the Clifford+*T*
base. From Amy et al. (http://arxiv.org/abs/1206.0758v3, Figure
5(b)). The first argument is the target, and the second one is the
control. The control can be positive or negative.

controlled_T_at :: Qubit -> Signed Qubit -> Circ () Source #

Decomposition of a controlled *T*-gate into the Clifford+*T*
base. The first argument is the target, and the second one is the
control. The control can be positive or negative.

controlled_V_at :: Qubit -> Signed Qubit -> Circ () Source #

Decomposition of a controlled *V*-gate into the Clifford+*T*
base. Adapted from Amy et al. (http://arxiv.org/abs/1206.0758v3,
Figure 5(c)). Our *V*-gate is *H**S*^{†}*H* as in Nielsen and
Chuang. The first argument is the target, and the second one is the
control. The control can be positive or negative.

controlled_E_at :: Qubit -> Signed Qubit -> Circ () Source #

Decomposition of a controlled *E*-gate into the Clifford+*T*
base. The first argument is the target, and the second one is the
control. The control can be positive or negative.

controlled_YY_at :: Qubit -> Signed Qubit -> Circ () Source #

Decomposition of a controlled **Y**-gate into the Clifford+*T*
base. The gate is from the Ground State Estimation algorithm and is
defined as **Y** = *SHS*, or equivalently,

It should not be confused with the Pauli *Y* gate.
The first argument is the target, and the second one is the
control. The control can be positive or negative.

toffoli_plain_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

A "plain" Toffoli gate, not decomposed. This is provided for
convenience, for example to use with `with_combined_controls`

.

cc_iX_plain_at :: Qubit -> Signed Qubit -> Signed Qubit -> Circ () Source #

A "plain" doubly-controlled *iX*-gate, not decomposed. This is
provided for convenience, for example to use with
`with_combined_controls`

.

multi_cnot_barenco_at :: (Qubit -> Signed Qubit -> Signed Qubit -> Circ ()) -> (Qubit -> Signed Qubit -> Signed Qubit -> Circ ()) -> Qubit -> [Qubit] -> [Signed Qubit] -> Circ () Source #

Decomposition of an *m*-times controlled not-gate, using *m*−2
ancillas that do not need be initialized in a particular
state. Adapted from Barenco et al.
(http://arxiv.org/abs/quantph/9503016, Lemma 7.2).

In addition to what is shown in Barenco et al., this function
permits some Toffoli gates to be replaced by doubly-controlled
*iX*-gates. This may be beneficial in gate bases, such as
Clifford+*T*, where a doubly-controlled *iX*-gate has a simpler
representation than a Toffoli gate.

The first argument is a Toffoli gate to use in the
decomposition. The second argument may be either a Toffoli gate or
a doubly-controlled *iX* gate. The third argument is the target,
the fourth argument is a list of qubits to be used as ancillas, and
the fifth argument is a list of signed controls. The ancillas need
not be initialized, and are returned in their original state.

The size of this circuit is linear in the number of controls; the
decomposition uses 4*m*−8 doubly-controlled gates for *m* ≥ 3.

multi_ciX_noancilla_at :: (Qubit -> Signed Qubit -> Signed Qubit -> Circ ()) -> Qubit -> [Signed Qubit] -> Circ () Source #

Decomposition of a multiply-controlled *iX*-gate, using no
ancillas. Adapted from Giles and Selinger
(http://arxiv.org/abs/1212.0506, Section 5.2).

The first argument is a Toffoli gate or a doubly-controlled
*iX*-gate. The third argument is the target, and the fourth
argument is a list of signed controls.

The size of this circuit is linear in the number of controls; the
decomposition uses 8*m*−32 doubly-controlled gates, 4 *T*-gates,
and 2 *H*-gates, for *m* ≥ 6.

# Decomposition of controls

partition_controls :: [Signed Endpoint] -> ([Signed Qubit], [Signed Bit]) Source #

Partition a list of controls into quantum and classical.

with_signed_qubit :: Signed Qubit -> (Qubit -> Circ b) -> Circ b Source #

Given a function that expects a qubit (typically as a control),
turn it into a function that can handle a *signed* (positive or
negative) qubit. This is done by conjugating the circuit with
negations on both sides, if the sign is negative. Usage:

with_signed_qubit c $ \q -> do <<<code using q>>>

with_combined_controls :: (Qubit -> Signed Qubit -> Signed Qubit -> Circ ()) -> Int -> [Signed Endpoint] -> ([Signed Qubit] -> Circ a) -> Circ a Source #

Decompose quantum controls recursively until at most *n* remain,
and then pass these reduced controls to the given circuit.
Precondition: *n* ≥ 1.

The decomposition is done using a Toffoli-like gate that is given
as the first argument. This should be either a Toffoli gate, a
doubly-controlled *iX*-gate, a decomposition thereof, or any other
reversible ternary gate with the behavior

- |000〉 ↦ |0〉|φ
_{0}〉 - |001〉 ↦ |0〉|φ
_{1}〉 - |010〉 ↦ |0〉|φ
_{2}〉 - |011〉 ↦ |1〉|φ
_{3}〉,

where the states |φ_{0}〉, …, |φ_{3}〉 are arbitrary.

For example, when *n*=2, this typically yields a circuit such as
the following (here shown using the doubly-controlled *iX*-gate):

And for *n*=1, the circuit typically looks like this:

Classical controls are not decomposed, but are applied to the resulting circuit directly.