Expression of type Implies ¶

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, Lambda, Variable, VertExprArray, l, m
from proveit.logic import And, Bijections, Implies, InSet
from proveit.numbers import Add, Interval, Mod, 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, _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 = 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))
expr = Implies(And(InSet(Lambda(l, Mod(Add(_b_floor, l), _two_pow_t)), Bijections(_full_domain, _m_domain)), InSet(Lambda(m, _phase_est_circuit), Bijections(_m_domain, _Omega))), InSet(Lambda(l, Conditional(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)])), InSet(l, _full_domain))), Bijections(_full_domain, _Omega))).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(\left(\left[l \mapsto \left(\left(b_{\textit{f}} + l\right) ~\textup{mod}~ 2^{t}\right)\right] \in \left[\{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\} \xrightarrow[\text{onto}]{\text{1-to-1}} \{0~\ldotp \ldotp~2^{t} - 1\}\right]\right) \land \left(\left[m \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 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)\right] \in \left[\{0~\ldotp \ldotp~2^{t} - 1\} \xrightarrow[\text{onto}]{\text{1-to-1}} \Omega\right]\right)\right) \\  \Rightarrow \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} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\}\right..\right] \in \left[\{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\} \xrightarrow[\text{onto}]{\text{1-to-1}} \Omega\right]\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	Operation	operator: 5 operands: 6
4	Operation	operator: 30 operands: 7
5	Literal
6	ExprTuple	8, 9
7	ExprTuple	10, 11
8	Operation	operator: 30 operands: 12
9	Operation	operator: 30 operands: 13
10	Lambda	parameter: 147 body: 14
11	Operation	operator: 27 operands: 15
12	ExprTuple	16, 17
13	ExprTuple	18, 19
14	Conditional	value: 20 condition: 21
15	ExprTuple	39, 37
16	Lambda	parameter: 147 body: 23
17	Operation	operator: 27 operands: 24
18	Lambda	parameter: 140 body: 26
19	Operation	operator: 27 operands: 28
20	Operation	operator: 34 operands: 29
21	Operation	operator: 30 operands: 31
22	ExprTuple	147
23	Operation	operator: 32 operands: 33
24	ExprTuple	39, 36
25	ExprTuple	140
26	Operation	operator: 34 operands: 35
27	Literal
28	ExprTuple	36, 37
29	ExprTuple	41, 42, 43, 38
30	Literal
31	ExprTuple	147, 39
32	Literal
33	ExprTuple	40, 78
34	Literal
35	ExprTuple	41, 42, 43, 44
36	Operation	operator: 115 operands: 45
37	Literal
38	ExprTuple	46, 55
39	Operation	operator: 115 operands: 47
40	Operation	operator: 137 operands: 143
41	ExprTuple	48, 49
42	ExprTuple	50, 51
43	ExprTuple	52, 53
44	ExprTuple	54, 55
45	ExprTuple	56, 57
46	ExprRange	lambda_map: 58 start_index: 150 end_index: 144
47	ExprTuple	59, 104
48	ExprRange	lambda_map: 60 start_index: 150 end_index: 144
49	ExprRange	lambda_map: 61 start_index: 150 end_index: 141
50	ExprRange	lambda_map: 62 start_index: 150 end_index: 144
51	ExprRange	lambda_map: 62 start_index: 125 end_index: 126
52	ExprRange	lambda_map: 63 start_index: 150 end_index: 144
53	ExprRange	lambda_map: 64 start_index: 150 end_index: 141
54	ExprRange	lambda_map: 65 start_index: 150 end_index: 144
55	ExprRange	lambda_map: 66 start_index: 150 end_index: 141
56	Literal
57	Operation	operator: 137 operands: 67
58	Lambda	parameter: 124 body: 68
59	Operation	operator: 137 operands: 69
60	Lambda	parameter: 124 body: 70
61	Lambda	parameter: 124 body: 71
62	Lambda	parameter: 124 body: 72
63	Lambda	parameter: 124 body: 73
64	Lambda	parameter: 124 body: 74
65	Lambda	parameter: 124 body: 75
66	Lambda	parameter: 124 body: 77
67	ExprTuple	78, 145
68	Operation	operator: 88 operands: 79
69	ExprTuple	80, 150
70	Operation	operator: 107 operands: 81
71	Operation	operator: 88 operands: 82
72	Operation	operator: 88 operands: 83
73	Operation	operator: 84 operands: 85
74	Operation	operator: 108 operands: 86
75	Operation	operator: 88 operands: 87
76	ExprTuple	124
77	Operation	operator: 88 operands: 89
78	Operation	operator: 118 operands: 90
79	NamedExprs	element: 91 targets: 100
80	Operation	operator: 148 operand: 104
81	NamedExprs	state: 93
82	NamedExprs	element: 94 targets: 102
83	NamedExprs	element: 95 targets: 96
84	Literal
85	NamedExprs	basis: 97
86	NamedExprs	operation: 98
87	NamedExprs	element: 99 targets: 100
88	Literal
89	NamedExprs	element: 101 targets: 102
90	ExprTuple	128, 144
91	Operation	operator: 113 operands: 103
92	ExprTuple	104
93	Operation	operator: 105 operand: 120
94	Operation	operator: 107 operands: 114
95	Operation	operator: 108 operands: 109
96	Operation	operator: 115 operands: 110
97	Literal
98	Literal
99	Operation	operator: 113 operands: 111
100	Operation	operator: 115 operands: 112
101	Operation	operator: 113 operands: 114
102	Operation	operator: 115 operands: 116
103	NamedExprs	state: 117 part: 124
104	Operation	operator: 118 operands: 119
105	Literal
106	ExprTuple	120
107	Literal
108	Literal
109	NamedExprs	operation: 121 part: 124
110	ExprTuple	150, 126
111	NamedExprs	state: 122 part: 124
112	ExprTuple	150, 144
113	Literal
114	NamedExprs	state: 123 part: 124
115	Literal
116	ExprTuple	125, 126
117	Operation	operator: 132 operands: 127
118	Literal
119	ExprTuple	128, 129
120	Literal
121	Operation	operator: 130 operands: 131
122	Operation	operator: 132 operands: 133
123	Literal
124	Variable
125	Operation	operator: 137 operands: 134
126	Operation	operator: 137 operands: 135
127	ExprTuple	136, 144
128	Literal
129	Operation	operator: 137 operands: 138
130	Literal
131	ExprTuple	139, 144
132	Literal
133	ExprTuple	140, 144
134	ExprTuple	144, 150
135	ExprTuple	144, 141
136	Operation	operator: 142 operands: 143
137	Literal
138	ExprTuple	144, 145
139	Literal
140	Variable
141	Literal
142	Literal
143	ExprTuple	146, 147
144	Literal
145	Operation	operator: 148 operand: 150
146	Literal
147	Variable
148	Literal
149	ExprTuple	150
150	Literal

Expression of type Implies¶

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

Expression of type Implies ¶

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