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, 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
from proveit.physics.quantum.QPE import _ket_u, _phase, _s
from proveit.physics.quantum.circuits import MultiQubitElem, Output

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Neg(t)
sub_expr3 = Add(sub_expr2, one)
sub_expr4 = frac(one, sqrt(two))
sub_expr5 = ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = Interval(Add(t, one), Add(t, _s))), one, _s).with_wrapping_at(2,6)
sub_expr6 = Output(state = ScalarMult(sub_expr4, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))))
expr = Equals([Output(state = ScalarMult(sub_expr4, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr3)), _phase)), ket1)))), ExprRange(sub_expr1, sub_expr6, Add(sub_expr2, two), zero).with_decreasing_order(), sub_expr5], [ExprRange(sub_expr1, sub_expr6, sub_expr3, zero).with_decreasing_order(), sub_expr5]).with_wrapping_at(2)

expr:

# 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(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 1\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \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)} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 3} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \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)} 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}\right) =  \\ \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \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)} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \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)} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \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)} 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}\right) \end{array} \end{array}

stored_expr.style_options()

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')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	ExprTuple	5, 6, 8
4	ExprTuple	7, 8
5	Operation	operator: 27 operands: 9
6	ExprRange	lambda_map: 11 start_index: 10 end_index: 55
7	ExprRange	lambda_map: 11 start_index: 78 end_index: 55
8	ExprRange	lambda_map: 12 start_index: 85 end_index: 47
9	NamedExprs	state: 13
10	Operation	operator: 81 operands: 14
11	Lambda	parameter: 86 body: 15
12	Lambda	parameter: 86 body: 16
13	Operation	operator: 51 operands: 17
14	ExprTuple	84, 79
15	Operation	operator: 27 operands: 18
16	Operation	operator: 19 operands: 20
17	ExprTuple	32, 21
18	NamedExprs	state: 22
19	Literal
20	NamedExprs	element: 23 targets: 24
21	Operation	operator: 39 operands: 25
22	Operation	operator: 51 operands: 26
23	Operation	operator: 27 operands: 28
24	Operation	operator: 29 operands: 30
25	ExprTuple	45, 31
26	ExprTuple	32, 33
27	Literal
28	NamedExprs	state: 34 part: 86
29	Literal
30	ExprTuple	35, 36
31	Operation	operator: 51 operands: 37
32	Operation	operator: 59 operands: 38
33	Operation	operator: 39 operands: 40
34	Literal
35	Operation	operator: 81 operands: 41
36	Operation	operator: 81 operands: 42
37	ExprTuple	43, 57
38	ExprTuple	85, 44
39	Literal
40	ExprTuple	45, 46
41	ExprTuple	89, 85
42	ExprTuple	89, 47
43	Operation	operator: 76 operands: 48
44	Operation	operator: 76 operands: 49
45	Operation	operator: 62 operand: 55
46	Operation	operator: 51 operands: 52
47	Literal
48	ExprTuple	65, 53
49	ExprTuple	79, 54
50	ExprTuple	55
51	Literal
52	ExprTuple	56, 57
53	Operation	operator: 68 operands: 58
54	Operation	operator: 59 operands: 60
55	Literal
56	Operation	operator: 76 operands: 61
57	Operation	operator: 62 operand: 85
58	ExprTuple	79, 71, 72, 64, 74
59	Literal
60	ExprTuple	85, 79
61	ExprTuple	65, 66
62	Literal
63	ExprTuple	85
64	Operation	operator: 76 operands: 67
65	Literal
66	Operation	operator: 68 operands: 69
67	ExprTuple	79, 70
68	Literal
69	ExprTuple	79, 71, 72, 73, 74
70	Operation	operator: 87 operand: 78
71	Literal
72	Literal
73	Operation	operator: 76 operands: 77
74	Literal
75	ExprTuple	78
76	Literal
77	ExprTuple	79, 80
78	Operation	operator: 81 operands: 82
79	Literal
80	Operation	operator: 87 operand: 86
81	Literal
82	ExprTuple	84, 85
83	ExprTuple	86
84	Operation	operator: 87 operand: 89
85	Literal
86	Variable
87	Literal
88	ExprTuple	89
89	Variable