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, TensorProd, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase, _psi_t_ket

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Equals(_psi_t_ket, TensorProd(ExprRange(sub_expr1, 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()))

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())

\lvert \psi_{t} \rangle = \left(\left(\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)\right) {\otimes}  \left(\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)\right) {\otimes}  \ldots {\otimes}  \left(\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)\right)\right)

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.	()	()	('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	Operation	operator: 5 operand: 22
4	Operation	operator: 6 operands: 7
5	Literal
6	Literal
7	ExprTuple	8
8	ExprRange	lambda_map: 9 start_index: 10 end_index: 31
9	Lambda	parameter: 54 body: 11
10	Operation	operator: 12 operands: 13
11	Operation	operator: 28 operands: 14
12	Literal
13	ExprTuple	15, 41
14	ExprTuple	16, 17
15	Operation	operator: 52 operand: 22
16	Operation	operator: 34 operands: 19
17	Operation	operator: 20 operands: 21
18	ExprTuple	22
19	ExprTuple	41, 23
20	Literal
21	ExprTuple	24, 25
22	Variable
23	Operation	operator: 48 operands: 26
24	Operation	operator: 37 operand: 31
25	Operation	operator: 28 operands: 29
26	ExprTuple	50, 30
27	ExprTuple	31
28	Literal
29	ExprTuple	32, 33
30	Operation	operator: 34 operands: 35
31	Literal
32	Operation	operator: 48 operands: 36
33	Operation	operator: 37 operand: 41
34	Literal
35	ExprTuple	41, 50
36	ExprTuple	39, 40
37	Literal
38	ExprTuple	41
39	Literal
40	Operation	operator: 42 operands: 43
41	Literal
42	Literal
43	ExprTuple	50, 44, 45, 46, 47
44	Literal
45	Literal
46	Operation	operator: 48 operands: 49
47	Literal
48	Literal
49	ExprTuple	50, 51
50	Literal
51	Operation	operator: 52 operand: 54
52	Literal
53	ExprTuple	54
54	Variable