Prelude Reference

For simplicity, the types of the following library functions are annotated according to a width global analysis and depth local analysis. Please note that type annotations will likely be different when analysing other metrics.

Qubit handling

qinit0 :: ![1] Qubit{0}

Initialize a qubit in the basis state 0.

qinit1 :: ![1] Qubit{0}

Initialize a qubit in the basis state 1.

qdiscard :: ![0](forall[0,0] d. Qubit{d} -o[1,0] ())

Discard a qubit q at depth d.

Parameter	Description
d	Maximum depth of the input qubit.

Argument	Type	Description
q	Qubit{d}	The qubit to discard.

meas :: ![0](forall[0,0] d. Qubit{d} -o[1,0] Bit{d+1})

Measure a qubit q at depth d.

Parameter	Description
d	Maximum depth of the input qubit.

Argument	Type	Description
q	Qubit{d}	The qubit to measure.

Returns: a bit at depth d+1 containing the result of the measurement.

Bit handling

cinit0 :: ![1] Bit{0}

Initialize a bit in the 0 state.

cinit1 :: ![1] Bit{0}

Initialize a bit in the 1 state.

cdiscard :: ![0](forall[0,0] d. Bit{d} -o[1,0] ())

Discard a bit b at depth d.

Parameter	Description
d	Maximum depth of the input bit.

Argument	Type	Description
b	Bit{d}	The bit to discard.

Single-qubit gates

hadamard :: ![0](forall[0,0] d. Qubit{d} -o[1,0] Qubit{d+1})

Apply the Hadamard gate to a qubit q at depth d.

Parameter	Description
d	Maximum depth of the input qubit.

Argument	Type	Description
q	Qubit{d}	The input qubit to Hadamard.

Returns: a qubit at depth d+1.

qnot :: ![0](forall[0,0] d. Qubit{d} -o[1,0] Qubit{d+1})

Apply the Pauli-X gate to a qubit q at depth d. This behaves like classical negation on the basis states.

Parameter	Description
d	Maximum depth of the input qubit.

Argument	Type	Description
q	Qubit{d}	The input qubit to Pauli-X.

Returns: a qubit at depth d+1.

pauliY :: ![0](forall[0,0] d. Qubit{d} -o[1,0] Qubit{d+1})

Apply the Pauli-Y gate to a qubit q at depth d.

Parameter	Description
d	Maximum depth of the input qubit.

Argument	Type	Description
q	Qubit{d}	The input qubit to Pauli-Y.

Returns: a qubit at depth d+1.

pauliZ :: ![0](forall[0,0] d. Qubit{d} -o[1,0] Qubit{d+1})

Apply the Pauli-Z gate to a qubit q at depth d.

Parameter	Description
d	Maximum depth of the input qubit.

Argument	Type	Description
q	Qubit{d}	The input qubit to Pauli-Z.

Returns: a qubit at depth d+1.

tgate :: ![0](forall[0,0] d. Qubit{d} -o[1,0] Qubit{d+1})

Apply the T gate to a qubit q at depth d.

Parameter	Description
d	Maximum depth of the input qubit.

Argument	Type	Description
q	Qubit{d}	The input qubit to the T gate.

Returns: a qubit at depth d+1.

Controlled gates

cnot :: ![0](forall[0,0] dctrl. forall[0,0] dtrgt. Qubit{dctrl} -o[1,0] Qubit{dtrgt} -o[2,1] (Qubit{max(dctrl, dtrgt) + 1}, Qubit{max(dctrl, dtrgt) + 1}))

Apply the controlled-not gate to a control qubit ctrl at depth dctrl and a target qubit trgt at depth dtrgt.

Parameter	Description
dctrl	Maximum depth of the control qubit.
dtrgt	Maximum depth of the target qubit.

Argument	Type	Description
ctrl	Qubit{dctrl}	The control qubit.
trgt	Qubit{dtrgt}	The target qubit.

Returns: the control and target qubits, both at depth max(dctrl, dtrgt) + 1.

cz :: ![0](forall[0,0] dctrl. forall[0,0] dtrgt. Qubit{dctrl} -o[1,0] Qubit{dtrgt} -o[2,1] (Qubit{max(dctrl, dtrgt) + 1}, Qubit{max(dctrl, dtrgt) + 1}))

Apply the controlled-Z gate to a control qubit ctrl at depth dctrl and a target qubit trgt at depth dtrgt.

Parameter	Description
dctrl	Maximum depth of the control qubit.
dtrgt	Maximum depth of the target qubit.

Argument	Type	Description
ctrl	Qubit{dctrl}	The control qubit.
trgt	Qubit{dtrgt}	The target qubit.

Returns: the control and target qubits, both at depth max(dctrl, dtrgt) + 1.

toffoli :: ![0](forall[0,0] dctrl1. forall[0,0] dctrl2. forall[0,0] dtrgt. Qubit{dctrl1} -o[1,0] Qubit{dctrl2} -o[2,1] Qubit{dtrgt} -o[3,2] (Qubit{max(dctrl1, dctrl2, dtrgt) + 1}, Qubit{max(dctrl1, dctrl2, dtrgt) + 1}, Qubit{max(dctrl1, dctrl2, dtrgt) + 1}))

Apply the Toffoli gate to two control qubits ctrl1 and ctrl2 at depth dctrl1 and dctrl2, respectively, and a target qubit trgt at depth dtrgt.

Parameter	Description
dctrl1	Maximum depth of the first control qubit.
dctrl2	Maximum depth of the second control qubit.
dtrgt	Maximum depth of the target qubit.

Argument	Type	Description
ctrl1	Qubit{dctrl1}	The first control qubit.
ctrl2	Qubit{dctrl2}	The second control qubit.
trgt	Qubit{dtrgt}	The target qubit.

Returns: the two controls and the target qubit, all at depth max(dctrl1, dctrl2, dtrgt) + 1.

ccnot :: ![0](forall[0,0] dctrl. forall[0,0] dtrgt. Bit{dctrl} -o[1,0] Qubit{dtrgt} -o[2,1] (Bit{max(dctrl, dtrgt) + 1}, Qubit{max(dctrl, dtrgt) + 1}))

Apply the classically-controlled not gate to a control bit ctrl at depth dctrl and a target qubit trgt at depth dtrgt.

Parameter	Description
dctrl	Maximum depth of the control bit.
dtrgt	Maximum depth of the target qubit.

Argument	Type	Description
ctrl	Bit{dctrl}	The control bit.
trgt	Qubit{dtrgt}	The target qubit.

Returns: the control bit and the target qubit, both at depth max(dctrl, dtrgt) + 1.

ccz :: ![0](forall[0,0] dctrl. forall[0,0] dtrgt. Bit{dctrl} -o[1,0] Qubit{dtrgt} -o[2,1] (Bit{max(dctrl, dtrgt) + 1}, Qubit{max(dctrl, dtrgt) + 1}))

Apply the classically-controlled Z gate to a control bit ctrl at depth dctrl and a target qubit trgt at depth dtrgt.

Parameter	Description
dctrl	Maximum depth of the control bit.
dtrgt	Maximum depth of the target qubit.

Argument	Type	Description
ctrl	Bit{dctrl}	The control bit.
trgt	Qubit{dtrgt}	The target qubit.

Returns: the control bit and the target qubit, both at depth max(dctrl, dtrgt) + 1.

Parametric gates

rgate :: ![0](forall[0,0] n. forall[0,0] d. Qubit{d} -o[1,0] Qubit{d+1})

Parametric rotation gate. Apply a rotation of 2π/2^n radians around the Z-axis to a qubit q at depth d.

Parameter	Description
n	Determines the angle of the rotation.
d	Maximum depth of the target qubit.

Argument	Type	Description
q	Qubit{d}	The qubit to rotate.

Returns: a qubit at depth d+1.

invrgate :: ![0](forall[0,0] n. forall[0,0] d. Qubit{d} -o[1,0] Qubit{d+1})

Parametric rotation gate. Inverse of rgate. Apply a rotation of -2π/2^n radians around the Z-axis to a qubit q at depth d.

Parameter	Description
n	Determines the angle of the rotation.
d	Maximum depth of the target qubit.

Argument	Type	Description
q	Qubit{d}	The qubit to rotate.

Returns: a qubit at depth d+1.

cr :: ![0](forall[0,0] n. forall[0,0] dctrl. forall[0,0] dtrgt. Qubit{dctrl} -o[1,0] Qubit{dtrgt} -o[2,1] (Qubit{max(dctrl, dtrgt) + 1}, Qubit{max(dctrl, dtrgt) + 1}))

Parametric controlled rotation gate. Apply a rotation of 2π/2^n radians around the Z-axis to a control qubit ctrl at depth dctrl and a target qubit trgt at depth dtrgt. Note that this gate is symmetrical.

Parameter	Description
dctrl	Maximum depth of the control qubit.
dtrgt	Maximum depth of the target qubit.

Argument	Type	Description
ctrl	Qubit{dctrl}	The control qubit.
trgt	Qubit{dtrgt}	The target qubit.

Returns: the control bit and the target qubit, both at depth max(dctrl, dtrgt) + 1.

invcr :: ![0](forall[0,0] n. forall[0,0] dctrl. forall[0,0] dtrgt. Qubit{dctrl} -o[1,0] Qubit{dtrgt} -o[2,1] (Qubit{max(dctrl, dtrgt) + 1}, Qubit{max(dctrl, dtrgt) + 1}))

Parametric controlled rotation gate. Inverse of cr. Apply a rotation of -2π/2^n radians around the Z-axis to a control qubit ctrl at depth dctrl and a target qubit trgt at depth dtrgt. Note that this gate is symmetrical.

Parameter	Description
dctrl	Maximum depth of the control qubit.
dtrgt	Maximum depth of the target qubit.

Argument	Type	Description
ctrl	Qubit{dctrl}	The control qubit.
trgt	Qubit{dtrgt}	The target qubit.

Returns: the control bit and the target qubit, both at depth max(dctrl, dtrgt) + 1.

mcnot :: ![0](forall[0,0] n. forall[0,0] dctrls. forall[0,0] dtrgt. List[i<n] Qubit{dctrls} -o[n,0] Qubit{dtrgt} -o[n+1, n] (List[i<n] Qubit{max(dctrls, dtrgt) + 1}, Qubit{max(dctrls, dtrgt) + 1}))

Multi-controlled not gate. Apply a controlled-not operation to a list ctrls of n control qubits at depth dctrls and a target qubit trgt at depth dtrgt.

Note: although each gate in this family counts as a single elementary gate, a physical implementation is not guaranteed to exist. Take this into account when reading the results of the tool, especially during gate count analysis.

Parameter	Description
n	Number of control qubits.
dctrls	Maximum depth of the control qubits.
dtrgt	Maximum depth of the target qubit.

Argument	Type	Description
ctrls	List[i<n] Qubit{dctrls}	The list of control qubits.
trgt	Qubit{dtrgt}	The target qubit.

Other

range :: ![0](forall[0,0] n. List[i<n] ())

Generate a list of unit values for iteration purposes.

Parameter	Description
n	The desired length of the list.

Returns: a list containing exactly n unit values.