Expression of type Equals ¶

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, 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 = 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())

\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}

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.	()	(1)	('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',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

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

Expression of type Equals¶

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

Expression of type Equals ¶

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