Safe Haskell | None |
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This module provides a representation of the single-qubit
Clifford+*T* operators, Matsumoto-Amano normal forms, and functions
for the exact synthesis of single-qubit Clifford+*T* operators.

Matsumoto-Amano normal forms and the Matsumoto-Amano exact synthesis algorithm are described in the paper:

- Ken Matsumoto, Kazuyuki Amano. Representation of Quantum Circuits with Clifford and π/8 Gates. http://arxiv.org/abs/0806.3834.

- data Gate
- class ToGates a where
- class FromGates a where
- from_gates :: [Gate] -> a

- invert_gates :: [Gate] -> [Gate]
- convert :: (ToGates a, FromGates b) => a -> b
- u2_X :: Ring a => U2 a
- u2_Y :: ComplexRing a => U2 a
- u2_Z :: Ring a => U2 a
- u2_H :: RootHalfRing a => U2 a
- u2_S :: ComplexRing a => U2 a
- u2_T :: OmegaRing a => U2 a
- u2_E :: (OmegaRing a, RootHalfRing a) => U2 a
- u2_W :: OmegaRing a => U2 a
- u2_of_gate :: (RootHalfRing a, ComplexRing a) => Gate -> U2 a
- so3_X :: Ring a => SO3 a
- so3_Y :: Ring a => SO3 a
- so3_Z :: Ring a => SO3 a
- so3_H :: Ring a => SO3 a
- so3_S :: Ring a => SO3 a
- so3_E :: Ring a => SO3 a
- so3_T :: RootHalfRing a => SO3 a
- so3_of_gate :: RootHalfRing a => Gate -> SO3 a
- so3_of_u2 :: (Adjoint a, ComplexRing a, RealPart a b, HalfRing b) => U2 a -> SO3 b
- so3_of_clifford :: (ToClifford a, Ring b) => a -> SO3 b
- clifford_of_so3 :: (Ring a, Eq a, Adjoint a) => SO3 a -> Clifford
- data NormalForm = NormalForm Syllables Clifford
- data Syllables
- normalform_append :: NormalForm -> Gate -> NormalForm
- nf_id :: NormalForm
- nf_mult :: ToGates b => NormalForm -> b -> NormalForm
- nf_inv :: ToGates a => a -> NormalForm
- normalize :: ToGates a => a -> NormalForm
- synthesis_bloch :: SO3 DRootTwo -> [Gate]
- synthesis_u2 :: U2 DOmega -> [Gate]
- normalform_pack :: NormalForm -> Integer
- normalform_unpack :: Integer -> NormalForm
- clifford_pack :: Clifford -> Integer
- clifford_unpack :: Integer -> Clifford

# Clifford+*T* interchange format

It is convenient to have a simple but exact "interchange
format" for operators in the single-qubit Clifford+*T*
group. Different operator representations can be converted to and
from this format.

Our format is simply a list of gates from *X*, *Y*, *Z*, *H*, *S*,
*T*, and *E* = *H**S*^{3}ω^{3}, with the obvious
interpretation as a matrix product. We also include the global
phase gate *W* = ω = *e*^{iπ/4}. The *W* gate is ignored when
converting to or from representations that cannot represent global
phase (such as the Bloch sphere representation).

An enumeration type to represent symbolic basic gates (*X*, *Y*,
*Z*, *H*, *S*, *T*, *W*, *E*).

Note: when we use a list of `Gate`

s to express a sequence of
operators, the operators are meant to be applied right-to-left,
i.e., as in the mathematical notation for matrix multiplication.
This is the opposite of the quantum circuit notation.

A type class for all things that can be exactly converted to a
list of gates. These are the exact representations of the
single-qubit Clifford+*T* group.

A type class for all things that a list of gates can be converted
to. For example, a list of gates can be converted to an element of
*U*(2) or an element of *SO*(3), using various (exact or
approximate) representations of the matrix entries.

from_gates :: [Gate] -> aSource

Convert a list of gates to any suitable type.

FromGates Integer | |

FromGates String | |

FromGates NormalForm | |

FromGates [Gate] | |

RootHalfRing a => FromGates (SO3 a) | |

(RootHalfRing a, ComplexRing a) => FromGates (U2 a) |

invert_gates :: [Gate] -> [Gate]Source

Invert a gate list.

# Matrices in *U*(2) and *SO*(3)

## Matrices in *U*(2)

u2_Y :: ComplexRing a => U2 aSource

The Pauli *Y* operator.

u2_H :: RootHalfRing a => U2 aSource

The Hadamard operator.

u2_S :: ComplexRing a => U2 aSource

The *S* operator.

u2_E :: (OmegaRing a, RootHalfRing a) => U2 aSource

The *E* operator.

u2_of_gate :: (RootHalfRing a, ComplexRing a) => Gate -> U2 aSource

Convert a symbolic gate to the corresponding operator.

## Matrices in *SO*(3)

This is the Bloch sphere representation of single qubit operators.

so3_T :: RootHalfRing a => SO3 aSource

The *T* operator.

so3_of_gate :: RootHalfRing a => Gate -> SO3 aSource

Convert a symbolic gate to the corresponding Bloch sphere operator.

## Conversions

so3_of_u2 :: (Adjoint a, ComplexRing a, RealPart a b, HalfRing b) => U2 a -> SO3 bSource

Conversion from *U*(2) to *SO*(3).

so3_of_clifford :: (ToClifford a, Ring b) => a -> SO3 bSource

Convert a Clifford operator to a matrix in *SO*(3).

clifford_of_so3 :: (Ring a, Eq a, Adjoint a) => SO3 a -> CliffordSource

Convert a matrix in *SO*(3) to a Clifford gate. Throw an error if
the matrix isn't Clifford.

# Matsumoto-Amano normal forms

A Matsumoto-Amano normal form is a sequence of Clifford+*T*
operators that is of the form

- (ε |
*T*) (*HT*|*SHT*)^{*}*C*.

Here, ε is the empty sequence, *C* is any Clifford operator, and
the meanings of `"|"`

and `"*"`

are as for regular
expressions. Every single-qubit Clifford+*T* operator has a unique
Matsumoto-Amano normal form.

## Representation of normal forms

data NormalForm Source

A representation of normal forms, optimized for right multiplication.

Syllables is a circuit of the form (ε|*T*) (*HT*|*SHT*)^{*}.

normalform_append :: NormalForm -> Gate -> NormalFormSource

Right-multiply the given normal form by a gate.

## Group operations on normal forms

The identity as a normal form.

nf_mult :: ToGates b => NormalForm -> b -> NormalFormSource

Multiply two normal forms. The right factor can be any
`ToGates`

.

nf_inv :: ToGates a => a -> NormalFormSource

Invert a normal form. The input can be any `ToGates`

.

## Conversion to normal form

normalize :: ToGates a => a -> NormalFormSource

Convert any `ToGates`

list to a `NormalForm`

, thereby normalizing it.

# Exact synthesis

## Synthesis from *SO*(3)

synthesis_bloch :: SO3 DRootTwo -> [Gate]Source

Input an exact matrix in *SO*(3), and output the corresponding
Clifford+*T* normal form. It is an error if the given matrix is not
an element of *SO*(3), i.e., orthogonal with determinant 1.

This implementation uses the Matsumoto-Amano algorithm.

Note: the list of gates will be returned in right-to-left order, i.e., as in the mathematical notation for matrix multiplication. This is the opposite of the quantum circuit notation.

## Synthesis from *U*(2)

synthesis_u2 :: U2 DOmega -> [Gate]Source

Input an exact matrix in *U*(2), and output the corresponding
Clifford+*T* normal form. The behavior is undefined if the given
matrix is not an element of *U*(2), i.e., unitary with determinant
1.

We use a variant of the Kliuchnikov-Maslov-Mosca algorithm, as implemented in Quantum.Synthesis.MultiQubitSynthesis.

Note: the list of gates will be returned in right-to-left order, i.e., as in the mathematical notation for matrix multiplication. This is the opposite of the quantum circuit notation.

# Compact representation of normal forms

It is sometimes useful to store Clifford+*T* operators in a file;
for this purpose, we provide a very succinct encoding of
Clifford+*T* operators as bit strings, which are in turns
represented as integers.

Our bitwise encoding is as follows. The first regular expression represents the set of Matsumoto-Amano normal forms (with a particular presentation of the rightmost Clifford operator). The second regular expression, which has the same form, defines the corresponding bit string encoding.

- (ε|
*T*) (*HT*|*SHT*)^{*}(ε|*H*|*SH*) (ε|*X*) (ε|*S²*) (ε|*S*) (ε|ω⁴) (ε|ω²) (ε|ω) - (10|11) (0|1)
^{*}(00|01|10) (0|1) (0|1) (0|1) (0|1) (0|1) (0|1)

As a special case, the leading bits 10 are omitted in case the encoded operator is a Clifford operator. This ensures that the encoding of a Clifford operator is an integer from 0 to 191.

This format has the property that the encoded Clifford+*T*
operator can, in principle, be read off directly from the hexadecimal
representation of the bit string, with the following decoding:

Leftmost one or two hexadecimal digits:

0 = n/a 4 = HT 8 = HTHT c = THTHT 1 = see below 5 = SHT 9 = HTSHT d = THTSHT 2 = ε 6 = THT a = SHTHT e = TSHTHT 3 = T 7 = TSHT b = SHTSHT f = TSHTSHT 10 = HTHTHT 14 = SHTHTHT 18 = THTHTHT 1c = TSHTHTHT 11 = HTHTSHT 15 = SHTHTSHT 19 = THTHTSHT 1d = TSHTHTSHT 12 = HTSHTHT 16 = SHTSHTHT 1a = THTSHTHT 1e = TSHTSHTHT 13 = HTSHTSHT 17 = SHTSHTSHT 1b = THTSHTSHT 1f = TSHTSHTSHT

Central hexadecimal digit:

0 = HTHTHTHT 4 = HTSHTHTHT 8 = SHTHTHTHT c = SHTSHTHTHT 1 = HTHTHTSHT 5 = HTSHTHTSHT 9 = SHTHTHTSHT d = SHTSHTHTSHT 2 = HTHTSHTHT 6 = HTSHTSHTHT a = SHTHTSHTHT e = SHTSHTSHTHT 3 = HTHTSHTSHT 7 = HTSHTSHTSHT b = SHTHTSHTSHT f = SHTSHTSHTSHT

Second-to-rightmost hexadecimal digit:

0 = ε 4 = H 8 = SH c = n/a 1 = SS 5 = HSS 9 = SHSS d = n/a 2 = X 6 = HX a = SHX e = n/a 3 = XSS 7 = HXSS b = SHXSS f = n/a

Rightmost hexadecimal digit:

0 = ε 4 = ω⁴ 8 = S c = Sω⁴ 1 = ω 5 = ω⁵ 9 = Sω d = Sω⁵ 2 = ω² 6 = ω⁶ a = Sω² e = Sω⁶ 3 = ω³ 7 = ω⁷ b = Sω³ f = Sω⁷

For example, the hexadecimal integer

6bf723e31

encodes the Clifford+*T* operator

THT SHTHTSHTSHT SHTSHTSHTSHT HTSHTSHTSHT HTHTSHTHT HTHTSHTSHT SHTSHTSHTHT XSS ω.

normalform_pack :: NormalForm -> IntegerSource

Compactly encode a `NormalForm`

as an `Integer`

.

normalform_unpack :: Integer -> NormalFormSource

Decode a `NormalForm`

from its `Integer`

encoding. This is the
inverse of `normalform_pack`

.

clifford_pack :: Clifford -> IntegerSource

Encode a Clifford operator as an integer in the range 0−191.

clifford_unpack :: Integer -> CliffordSource

Decode a Clifford operator from its integer encoding. This is the
inverse of `clifford_pack`