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.

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 = HS³ω³, 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).

This is the Bloch sphere representation of single qubit operators.

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.

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.