Expression of type Forall ¶

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 Conditional, ExprRange, Variable, VertExprArray, e, l
from proveit.logic import And, Equals, Forall, InSet, Union
from proveit.numbers import Add, Interval, ModAbs, greater, one
from proveit.physics.quantum import NumKet, Z, ket_plus
from proveit.physics.quantum.QPE import ModAdd, QPE, _U, _b_floor, _full_domain, _ket_u, _neg_domain, _pos_domain, _s, _s_wire, _t, _two_pow_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 = greater(ModAbs(l, _two_pow_t), e)
sub_expr7 = 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 = Forall(instance_param_or_params = [l], instance_expr = Equals(Conditional(sub_expr7, And(InSet(l, Union(_neg_domain, _pos_domain)), sub_expr6)), Conditional(sub_expr7, And(InSet(l, _full_domain), sub_expr6))).with_wrapping_at(1))

# 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())

\forall_{l}~\left(\begin{array}{c} \begin{array}{l} \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} \textrm{ if } l \in \left(\{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\} \cup \{e + 1~\ldotp \ldotp~2^{t - 1}\}\right) ,  \left|l\right|_{\textup{mod}\thinspace 2^{t}} > e\right.. \\  = \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} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\} ,  \left|l\right|_{\textup{mod}\thinspace 2^{t}} > e\right.. \end{array} \end{array}\right)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

	core type	sub-expressions
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 129 body: 5
4	ExprTuple	129
5	Operation	operator: 6 operands: 7
6	Literal
7	ExprTuple	8, 9
8	Conditional	value: 11 condition: 10
9	Conditional	value: 11 condition: 12
10	Operation	operator: 16 operands: 13
11	Operation	operator: 14 operands: 15
12	Operation	operator: 16 operands: 17
13	ExprTuple	18, 24
14	Literal
15	ExprTuple	19, 20, 21, 22
16	Literal
17	ExprTuple	23, 24
18	Operation	operator: 34 operands: 25
19	ExprTuple	26, 27
20	ExprTuple	28, 29
21	ExprTuple	30, 31
22	ExprTuple	32, 33
23	Operation	operator: 34 operands: 35
24	Operation	operator: 36 operands: 37
25	ExprTuple	129, 38
26	ExprRange	lambda_map: 39 start_index: 138 end_index: 134
27	ExprRange	lambda_map: 40 start_index: 138 end_index: 121
28	ExprRange	lambda_map: 41 start_index: 138 end_index: 134
29	ExprRange	lambda_map: 41 start_index: 107 end_index: 108
30	ExprRange	lambda_map: 42 start_index: 138 end_index: 134
31	ExprRange	lambda_map: 43 start_index: 138 end_index: 121
32	ExprRange	lambda_map: 44 start_index: 138 end_index: 134
33	ExprRange	lambda_map: 45 start_index: 138 end_index: 121
34	Literal
35	ExprTuple	129, 46
36	Literal
37	ExprTuple	123, 47
38	Operation	operator: 48 operands: 49
39	Lambda	parameter: 106 body: 50
40	Lambda	parameter: 106 body: 51
41	Lambda	parameter: 106 body: 52
42	Lambda	parameter: 106 body: 53
43	Lambda	parameter: 106 body: 54
44	Lambda	parameter: 106 body: 55
45	Lambda	parameter: 106 body: 57
46	Operation	operator: 98 operands: 58
47	Operation	operator: 59 operands: 60
48	Literal
49	ExprTuple	61, 62
50	Operation	operator: 90 operands: 63
51	Operation	operator: 70 operands: 64
52	Operation	operator: 70 operands: 65
53	Operation	operator: 66 operands: 67
54	Operation	operator: 91 operands: 68
55	Operation	operator: 70 operands: 69
56	ExprTuple	106
57	Operation	operator: 70 operands: 71
58	ExprTuple	86, 122
59	Literal
60	ExprTuple	129, 72
61	Operation	operator: 98 operands: 73
62	Operation	operator: 98 operands: 74
63	NamedExprs	state: 75
64	NamedExprs	element: 76 targets: 84
65	NamedExprs	element: 77 targets: 78
66	Literal
67	NamedExprs	basis: 79
68	NamedExprs	operation: 80
69	NamedExprs	element: 81 targets: 82
70	Literal
71	NamedExprs	element: 83 targets: 84
72	Operation	operator: 126 operands: 85
73	ExprTuple	86, 87
74	ExprTuple	110, 122
75	Operation	operator: 88 operand: 102
76	Operation	operator: 90 operands: 97
77	Operation	operator: 91 operands: 92
78	Operation	operator: 98 operands: 93
79	Literal
80	Literal
81	Operation	operator: 96 operands: 94
82	Operation	operator: 98 operands: 95
83	Operation	operator: 96 operands: 97
84	Operation	operator: 98 operands: 99
85	ExprTuple	130, 134
86	Operation	operator: 132 operands: 100
87	Operation	operator: 136 operand: 110
88	Literal
89	ExprTuple	102
90	Literal
91	Literal
92	NamedExprs	operation: 103 part: 106
93	ExprTuple	138, 108
94	NamedExprs	state: 104 part: 106
95	ExprTuple	138, 134
96	Literal
97	NamedExprs	state: 105 part: 106
98	Literal
99	ExprTuple	107, 108
100	ExprTuple	109, 138
101	ExprTuple	110
102	Literal
103	Operation	operator: 111 operands: 112
104	Operation	operator: 113 operands: 114
105	Literal
106	Variable
107	Operation	operator: 132 operands: 115
108	Operation	operator: 132 operands: 116
109	Operation	operator: 136 operand: 122
110	Operation	operator: 132 operands: 118
111	Literal
112	ExprTuple	119, 134
113	Literal
114	ExprTuple	120, 134
115	ExprTuple	134, 138
116	ExprTuple	134, 121
117	ExprTuple	122
118	ExprTuple	123, 138
119	Literal
120	Operation	operator: 124 operands: 125
121	Literal
122	Operation	operator: 126 operands: 127
123	Variable
124	Literal
125	ExprTuple	128, 129
126	Literal
127	ExprTuple	130, 131
128	Literal
129	Variable
130	Literal
131	Operation	operator: 132 operands: 133
132	Literal
133	ExprTuple	134, 135
134	Literal
135	Operation	operator: 136 operand: 138
136	Literal
137	ExprTuple	138
138	Literal

Expression of type Forall¶

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

Expression of type Forall ¶

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