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, t
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import ket0, ket1, ket_plus
from proveit.physics.quantum.QPE import QPE1, _U, _ket_u, _phase, _s
from proveit.physics.quantum.circuits import Gate, Input, 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 = QPE1(_U, t), part = sub_expr1), targets = Interval(one, sub_expr3))
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, Output(state = ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))), Add(Neg(t), one), zero).with_decreasing_order(), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]))), one)

# 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}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\vdots} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\lvert u \rangle}
} \end{array}\right) = 1

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

Expression of type Equals¶

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

Expression of type Equals ¶

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