Expression of type Less ¶

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, U, Variable, VertExprArray, s, t
from proveit.numbers import Add, Exp, Interval, Mod, Mult, Round, four, frac, greater, one, pi, two
from proveit.physics.quantum import I, NumKet, Z, ket_plus, var_ket_u
from proveit.physics.quantum.QPE import QPE, phase, 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))
expr = greater(Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = var_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), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, s)], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(Mod(Round(Mult(two_pow_t, phase)), two_pow_t), t), part = sub_expr1), targets = Interval(one, t)), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = var_ket_u, part = sub_expr1), targets = sub_expr4), one, s)]))), frac(four, Exp(pi, 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) > \frac{4}{\pi^{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	reversed	('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: 7 operand: 11
5	Literal
6	ExprTuple	9, 10
7	Literal
8	ExprTuple	11
9	Literal
10	Operation	operator: 102 operands: 12
11	Operation	operator: 13 operands: 14
12	ExprTuple	15, 104
13	Literal
14	ExprTuple	16, 17, 18, 19
15	Literal
16	ExprTuple	20, 21
17	ExprTuple	22, 23
18	ExprTuple	24, 25
19	ExprTuple	26, 27
20	ExprRange	lambda_map: 28 start_index: 90 end_index: 105
21	ExprRange	lambda_map: 29 start_index: 90 end_index: 91
22	ExprRange	lambda_map: 30 start_index: 90 end_index: 105
23	ExprRange	lambda_map: 30 start_index: 79 end_index: 80
24	ExprRange	lambda_map: 31 start_index: 90 end_index: 105
25	ExprRange	lambda_map: 32 start_index: 90 end_index: 91
26	ExprRange	lambda_map: 33 start_index: 90 end_index: 105
27	ExprRange	lambda_map: 34 start_index: 90 end_index: 91
28	Lambda	parameter: 78 body: 35
29	Lambda	parameter: 78 body: 36
30	Lambda	parameter: 78 body: 37
31	Lambda	parameter: 78 body: 38
32	Lambda	parameter: 78 body: 39
33	Lambda	parameter: 78 body: 40
34	Lambda	parameter: 78 body: 42
35	Operation	operator: 64 operands: 43
36	Operation	operator: 50 operands: 44
37	Operation	operator: 50 operands: 45
38	Operation	operator: 46 operands: 47
39	Operation	operator: 65 operands: 48
40	Operation	operator: 50 operands: 49
41	ExprTuple	78
42	Operation	operator: 50 operands: 51
43	NamedExprs	state: 52
44	NamedExprs	element: 53 targets: 61
45	NamedExprs	element: 54 targets: 55
46	Literal
47	NamedExprs	basis: 56
48	NamedExprs	operation: 57
49	NamedExprs	element: 58 targets: 59
50	Literal
51	NamedExprs	element: 60 targets: 61
52	Operation	operator: 62 operand: 74
53	Operation	operator: 64 operands: 71
54	Operation	operator: 65 operands: 66
55	Operation	operator: 72 operands: 67
56	Literal
57	Literal
58	Operation	operator: 70 operands: 68
59	Operation	operator: 72 operands: 69
60	Operation	operator: 70 operands: 71
61	Operation	operator: 72 operands: 73
62	Literal
63	ExprTuple	74
64	Literal
65	Literal
66	NamedExprs	operation: 75 part: 78
67	ExprTuple	90, 80
68	NamedExprs	state: 76 part: 78
69	ExprTuple	90, 105
70	Literal
71	NamedExprs	state: 77 part: 78
72	Literal
73	ExprTuple	79, 80
74	Literal
75	Operation	operator: 81 operands: 82
76	Operation	operator: 83 operands: 84
77	Variable
78	Variable
79	Operation	operator: 86 operands: 85
80	Operation	operator: 86 operands: 87
81	Literal
82	ExprTuple	88, 105
83	Literal
84	ExprTuple	89, 105
85	ExprTuple	105, 90
86	Literal
87	ExprTuple	105, 91
88	Variable
89	Operation	operator: 92 operands: 93
90	Literal
91	Variable
92	Literal
93	ExprTuple	94, 100
94	Operation	operator: 95 operand: 97
95	Literal
96	ExprTuple	97
97	Operation	operator: 98 operands: 99
98	Literal
99	ExprTuple	100, 101
100	Operation	operator: 102 operands: 103
101	Variable
102	Literal
103	ExprTuple	104, 105
104	Literal
105	Variable

Expression of type Less¶

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

Expression of type Less ¶

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