Expression of type SubsetEq ¶

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, IndexedVar, Variable, VertExprArray, e, m
from proveit.logic import SetOfAll, SubsetEq
from proveit.numbers import Add, Interval, ModAbs, greater, one, subtract
from proveit.physics.quantum import NumKet, Z, ket_plus
from proveit.physics.quantum.QPE import QPE, _U, _b_floor, _ket_u, _m_domain, _phase_est_circuit, _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 = IndexedVar(m, one)
sub_expr3 = Add(_t, one)
sub_expr4 = Add(_t, _s)
sub_expr5 = Interval(sub_expr3, sub_expr4)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr4))
expr = SubsetEq(SetOfAll(instance_param_or_params = [m], instance_element = _phase_est_circuit, domain = _m_domain, condition = greater(ModAbs(subtract(m, _b_floor), _two_pow_t), e)), SetOfAll(instance_param_or_params = [sub_expr2], instance_element = 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_expr5), one, _s)], [ExprRange(sub_expr1, sub_expr6, one, _t), ExprRange(sub_expr1, sub_expr6, sub_expr3, sub_expr4)], [ExprRange(sub_expr1, Measure(basis = Z), one, _t), _s_wire], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(sub_expr2, _t), part = sub_expr1), targets = Interval(one, _t)), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)])), domain = _m_domain)).with_wrapping_at(2)

# 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\{\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \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 m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right)~|~\left|m - b_{\textit{f}}\right|_{\textup{mod}\thinspace 2^{t}} > e\right\}_{m \in \{0~\ldotp \ldotp~2^{t} - 1\}} \subseteq  \\ \left\{\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert m_{1} \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m_{1} \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 m_{1} \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m_{1} \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right\}_{m_{1} \in \{0~\ldotp \ldotp~2^{t} - 1\}} \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.	()	(2)	('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	Operation	operator: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameter: 135 body: 11
9	Lambda	parameter: 132 body: 13
10	ExprTuple	135
11	Conditional	value: 14 condition: 15
12	ExprTuple	132
13	Conditional	value: 16 condition: 17
14	Operation	operator: 21 operands: 18
15	Operation	operator: 19 operands: 20
16	Operation	operator: 21 operands: 22
17	Operation	operator: 32 operands: 23
18	ExprTuple	27, 28, 29, 24
19	Literal
20	ExprTuple	25, 26
21	Literal
22	ExprTuple	27, 28, 29, 30
23	ExprTuple	132, 45
24	ExprTuple	31, 43
25	Operation	operator: 32 operands: 33
26	Operation	operator: 34 operands: 35
27	ExprTuple	36, 37
28	ExprTuple	38, 39
29	ExprTuple	40, 41
30	ExprTuple	42, 43
31	ExprRange	lambda_map: 44 start_index: 137 end_index: 133
32	Literal
33	ExprTuple	135, 45
34	Literal
35	ExprTuple	46, 47
36	ExprRange	lambda_map: 48 start_index: 137 end_index: 133
37	ExprRange	lambda_map: 49 start_index: 137 end_index: 134
38	ExprRange	lambda_map: 50 start_index: 137 end_index: 133
39	ExprRange	lambda_map: 50 start_index: 119 end_index: 120
40	ExprRange	lambda_map: 51 start_index: 137 end_index: 133
41	ExprRange	lambda_map: 52 start_index: 137 end_index: 134
42	ExprRange	lambda_map: 53 start_index: 137 end_index: 133
43	ExprRange	lambda_map: 54 start_index: 137 end_index: 134
44	Lambda	parameter: 118 body: 55
45	Operation	operator: 107 operands: 56
46	Variable
47	Operation	operator: 57 operands: 58
48	Lambda	parameter: 118 body: 59
49	Lambda	parameter: 118 body: 60
50	Lambda	parameter: 118 body: 61
51	Lambda	parameter: 118 body: 62
52	Lambda	parameter: 118 body: 63
53	Lambda	parameter: 118 body: 64
54	Lambda	parameter: 118 body: 66
55	Operation	operator: 78 operands: 67
56	ExprTuple	68, 69
57	Literal
58	ExprTuple	70, 94
59	Operation	operator: 99 operands: 71
60	Operation	operator: 78 operands: 72
61	Operation	operator: 78 operands: 73
62	Operation	operator: 74 operands: 75
63	Operation	operator: 100 operands: 76
64	Operation	operator: 78 operands: 77
65	ExprTuple	118
66	Operation	operator: 78 operands: 79
67	NamedExprs	element: 80 targets: 90
68	Literal
69	Operation	operator: 129 operands: 81
70	Operation	operator: 129 operands: 82
71	NamedExprs	state: 83
72	NamedExprs	element: 84 targets: 92
73	NamedExprs	element: 85 targets: 86
74	Literal
75	NamedExprs	basis: 87
76	NamedExprs	operation: 88
77	NamedExprs	element: 89 targets: 90
78	Literal
79	NamedExprs	element: 91 targets: 92
80	Operation	operator: 105 operands: 93
81	ExprTuple	94, 95
82	ExprTuple	135, 96
83	Operation	operator: 97 operand: 114
84	Operation	operator: 99 operands: 106
85	Operation	operator: 100 operands: 101
86	Operation	operator: 107 operands: 102
87	Literal
88	Literal
89	Operation	operator: 105 operands: 103
90	Operation	operator: 107 operands: 104
91	Operation	operator: 105 operands: 106
92	Operation	operator: 107 operands: 108
93	NamedExprs	state: 109 part: 118
94	Operation	operator: 110 operands: 111
95	Operation	operator: 112 operand: 137
96	Operation	operator: 112 operand: 123
97	Literal
98	ExprTuple	114
99	Literal
100	Literal
101	NamedExprs	operation: 115 part: 118
102	ExprTuple	137, 120
103	NamedExprs	state: 116 part: 118
104	ExprTuple	137, 133
105	Literal
106	NamedExprs	state: 117 part: 118
107	Literal
108	ExprTuple	119, 120
109	Operation	operator: 126 operands: 121
110	Literal
111	ExprTuple	122, 133
112	Literal
113	ExprTuple	123
114	Literal
115	Operation	operator: 124 operands: 125
116	Operation	operator: 126 operands: 127
117	Literal
118	Variable
119	Operation	operator: 129 operands: 128
120	Operation	operator: 129 operands: 130
121	ExprTuple	135, 133
122	Literal
123	Literal
124	Literal
125	ExprTuple	131, 133
126	Literal
127	ExprTuple	132, 133
128	ExprTuple	133, 137
129	Literal
130	ExprTuple	133, 134
131	Literal
132	IndexedVar	variable: 135 index: 137
133	Literal
134	Literal
135	Variable
136	ExprTuple	137
137	Literal

Expression of type SubsetEq¶

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

Expression of type SubsetEq ¶

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