Expression of type And ¶

from the theory of proveit.physics.quantum.QPE ¶

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import ExprRange, Lambda, Variable, VertExprArray, l
from proveit.logic import And, InSet, Injections, Surjections
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum import NumKet, Z, ket_plus
from proveit.physics.quantum.QPE import ModAdd, QPE, _Omega, _U, _b_floor, _full_domain, _ket_u, _s, _s_wire, _t
from proveit.physics.quantum.circuits import Gate, Input, Measure, MultiQubitElem, Output, Qcircuit

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(_t, one)
sub_expr3 = Add(_t, _s)
sub_expr4 = Interval(sub_expr2, sub_expr3)
sub_expr5 = MultiQubitElem(element = Gate(operation = QPE(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr3))
sub_expr6 = Lambda(l, Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)], [ExprRange(sub_expr1, sub_expr5, one, _t), ExprRange(sub_expr1, sub_expr5, sub_expr2, sub_expr3)], [ExprRange(sub_expr1, Measure(basis = Z), one, _t), _s_wire], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(ModAdd(_b_floor, l), _t), part = sub_expr1), targets = Interval(one, _t)), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)])))
expr = And(InSet(sub_expr6, Injections(_full_domain, _Omega)), InSet(sub_expr6, Surjections(_full_domain, _Omega)))

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(\left[l \mapsto \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right)\right] \in \left[\{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\} \xrightarrow[]{\text{1-to-1}} \Omega\right]\right) \land \left(\left[l \mapsto \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right)\right] \in \left[\{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\} \xrightarrow[\text{onto}]{} \Omega\right]\right)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)

	core type	sub-expressions
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	9, 8
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 11 operands: 15
9	Lambda	parameter: 113 body: 13
10	Operation	operator: 14 operands: 15
11	Literal
12	ExprTuple	113
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	20, 21, 22, 23
18	Operation	operator: 82 operands: 24
19	Literal
20	ExprTuple	25, 26
21	ExprTuple	27, 28
22	ExprTuple	29, 30
23	ExprTuple	31, 32
24	ExprTuple	33, 71
25	ExprRange	lambda_map: 34 start_index: 114 end_index: 106
26	ExprRange	lambda_map: 35 start_index: 114 end_index: 105
27	ExprRange	lambda_map: 36 start_index: 114 end_index: 106
28	ExprRange	lambda_map: 36 start_index: 91 end_index: 92
29	ExprRange	lambda_map: 37 start_index: 114 end_index: 106
30	ExprRange	lambda_map: 38 start_index: 114 end_index: 105
31	ExprRange	lambda_map: 39 start_index: 114 end_index: 106
32	ExprRange	lambda_map: 40 start_index: 114 end_index: 105
33	Operation	operator: 101 operands: 41
34	Lambda	parameter: 90 body: 42
35	Lambda	parameter: 90 body: 43
36	Lambda	parameter: 90 body: 44
37	Lambda	parameter: 90 body: 45
38	Lambda	parameter: 90 body: 46
39	Lambda	parameter: 90 body: 47
40	Lambda	parameter: 90 body: 49
41	ExprTuple	50, 114
42	Operation	operator: 74 operands: 51
43	Operation	operator: 58 operands: 52
44	Operation	operator: 58 operands: 53
45	Operation	operator: 54 operands: 55
46	Operation	operator: 75 operands: 56
47	Operation	operator: 58 operands: 57
48	ExprTuple	90
49	Operation	operator: 58 operands: 59
50	Operation	operator: 110 operand: 71
51	NamedExprs	state: 61
52	NamedExprs	element: 62 targets: 70
53	NamedExprs	element: 63 targets: 64
54	Literal
55	NamedExprs	basis: 65
56	NamedExprs	operation: 66
57	NamedExprs	element: 67 targets: 68
58	Literal
59	NamedExprs	element: 69 targets: 70
60	ExprTuple	71
61	Operation	operator: 72 operand: 86
62	Operation	operator: 74 operands: 81
63	Operation	operator: 75 operands: 76
64	Operation	operator: 82 operands: 77
65	Literal
66	Literal
67	Operation	operator: 80 operands: 78
68	Operation	operator: 82 operands: 79
69	Operation	operator: 80 operands: 81
70	Operation	operator: 82 operands: 83
71	Operation	operator: 84 operands: 85
72	Literal
73	ExprTuple	86
74	Literal
75	Literal
76	NamedExprs	operation: 87 part: 90
77	ExprTuple	114, 92
78	NamedExprs	state: 88 part: 90
79	ExprTuple	114, 106
80	Literal
81	NamedExprs	state: 89 part: 90
82	Literal
83	ExprTuple	91, 92
84	Literal
85	ExprTuple	93, 94
86	Literal
87	Operation	operator: 95 operands: 96
88	Operation	operator: 97 operands: 98
89	Literal
90	Variable
91	Operation	operator: 101 operands: 99
92	Operation	operator: 101 operands: 100
93	Literal
94	Operation	operator: 101 operands: 102
95	Literal
96	ExprTuple	103, 106
97	Literal
98	ExprTuple	104, 106
99	ExprTuple	106, 114
100	ExprTuple	106, 105
101	Literal
102	ExprTuple	106, 107
103	Literal
104	Operation	operator: 108 operands: 109
105	Literal
106	Literal
107	Operation	operator: 110 operand: 114
108	Literal
109	ExprTuple	112, 113
110	Literal
111	ExprTuple	114
112	Literal
113	Variable
114	Literal

Expression of type And¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type And ¶

from the theory of proveit.physics.quantum.QPE ¶