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 ExprRange, Variable, VertExprArray
from proveit.logic import Equals
from proveit.numbers import Abs, Add, Exp, Interval, Mod, Mult, Round, one, two
from proveit.physics.quantum import NumKet, Z, ket_plus
from proveit.physics.quantum.QPE import QPE, SubIndexed, _U, _alpha, _ket_u, _phase, _s, _s_wire, _t, _two_pow_t
from proveit.physics.quantum.circuits import Gate, Input, Measure, MultiQubitElem, Output, Qcircuit
from proveit.statistics import Prob

# 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 = Mod(Round(Mult(_two_pow_t, _phase)), _two_pow_t)
expr = Equals(Prob(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(sub_expr6, _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)]))), Exp(Abs(SubIndexed(_alpha, [sub_expr6])), two))

# 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())

\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \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 round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = \left|\alpha_{round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t}}\right|^{2}

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