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 Add, Interval, one
from proveit.physics.quantum import I, ket_plus
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import QPE1, _Psi_ket, _U, _ket_u, _psi__t_ket, _s, _t
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(one, _t)
sub_expr5 = Interval(sub_expr2, sub_expr3)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE1(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr3))
sub_expr7 = ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)
sub_expr8 = [ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(_t), part = sub_expr1), targets = sub_expr4), one, _t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, _s)]
sub_expr9 = [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _Psi_ket, part = sub_expr1), targets = sub_expr4), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)]
expr = Equals(Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, _t), sub_expr7], [ExprRange(sub_expr1, sub_expr6, one, _t), ExprRange(sub_expr1, sub_expr6, sub_expr2, sub_expr3)], sub_expr8, sub_expr9))), Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _psi__t_ket, part = sub_expr1), targets = sub_expr4), one, _t), sub_expr7], sub_expr8, sub_expr9)))).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} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}_1\left(U, t\right)} & \multigate{3}{{\mathrm {FT}}^{\dag}_{t}} & \multiqout{3}{\lvert \Psi \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi_{t} \rangle} & \gate{{\mathrm {FT}}^{\dag}_{t}} & \qout{\lvert \Psi \rangle} \\
\qin{\lvert u \rangle} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\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.	()	(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	Operation	operator: 11 operands: 10
9	Operation	operator: 11 operands: 12
10	ExprTuple	13, 14, 16, 17
11	Literal
12	ExprTuple	15, 16, 17
13	ExprTuple	18, 22
14	ExprTuple	19, 20
15	ExprTuple	21, 22
16	ExprTuple	23, 24
17	ExprTuple	25, 26
18	ExprRange	lambda_map: 27 start_index: 96 end_index: 97
19	ExprRange	lambda_map: 28 start_index: 96 end_index: 97
20	ExprRange	lambda_map: 28 start_index: 85 end_index: 86
21	ExprRange	lambda_map: 29 start_index: 96 end_index: 97
22	ExprRange	lambda_map: 30 start_index: 96 end_index: 98
23	ExprRange	lambda_map: 31 start_index: 96 end_index: 97
24	ExprRange	lambda_map: 32 start_index: 96 end_index: 98
25	ExprRange	lambda_map: 33 start_index: 96 end_index: 97
26	ExprRange	lambda_map: 34 start_index: 96 end_index: 98
27	Lambda	parameter: 84 body: 35
28	Lambda	parameter: 84 body: 36
29	Lambda	parameter: 84 body: 37
30	Lambda	parameter: 84 body: 38
31	Lambda	parameter: 84 body: 39
32	Lambda	parameter: 84 body: 40
33	Lambda	parameter: 84 body: 41
34	Lambda	parameter: 84 body: 43
35	Operation	operator: 69 operands: 44
36	Operation	operator: 51 operands: 45
37	Operation	operator: 51 operands: 46
38	Operation	operator: 51 operands: 47
39	Operation	operator: 51 operands: 48
40	Operation	operator: 70 operands: 49
41	Operation	operator: 51 operands: 50
42	ExprTuple	84
43	Operation	operator: 51 operands: 52
44	NamedExprs	state: 53
45	NamedExprs	element: 54 targets: 55
46	NamedExprs	element: 56 targets: 61
47	NamedExprs	element: 57 targets: 63
48	NamedExprs	element: 58 targets: 61
49	NamedExprs	operation: 59
50	NamedExprs	element: 60 targets: 61
51	Literal
52	NamedExprs	element: 62 targets: 63
53	Operation	operator: 64 operand: 78
54	Operation	operator: 70 operands: 66
55	Operation	operator: 76 operands: 67
56	Operation	operator: 69 operands: 68
57	Operation	operator: 69 operands: 75
58	Operation	operator: 70 operands: 71
59	Literal
60	Operation	operator: 74 operands: 72
61	Operation	operator: 76 operands: 73
62	Operation	operator: 74 operands: 75
63	Operation	operator: 76 operands: 77
64	Literal
65	ExprTuple	78
66	NamedExprs	operation: 79 part: 84
67	ExprTuple	96, 86
68	NamedExprs	state: 80 part: 84
69	Literal
70	Literal
71	NamedExprs	operation: 81 part: 84
72	NamedExprs	state: 82 part: 84
73	ExprTuple	96, 97
74	Literal
75	NamedExprs	state: 83 part: 84
76	Literal
77	ExprTuple	85, 86
78	Literal
79	Operation	operator: 87 operands: 88
80	Operation	operator: 89 operand: 97
81	Operation	operator: 90 operand: 97
82	Literal
83	Literal
84	Variable
85	Operation	operator: 93 operands: 92
86	Operation	operator: 93 operands: 94
87	Literal
88	ExprTuple	95, 97
89	Literal
90	Literal
91	ExprTuple	97
92	ExprTuple	97, 96
93	Literal
94	ExprTuple	97, 98
95	Literal
96	Literal
97	Literal
98	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 ¶