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 k, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd, VecSum
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, e, frac, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = frac(t, two)
sub_expr2 = frac(one, Exp(two, sub_expr1))
sub_expr3 = frac(one, Exp(two, subtract(frac(Add(t, one), two), sub_expr1)))
sub_expr4 = VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1))
sub_expr5 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, t)), domain = Interval(zero, subtract(two_pow_t, one)))
expr = Equals(ScalarMult(sub_expr3, TensorProd(sub_expr4, ScalarMult(sub_expr2, sub_expr5))), ScalarMult(sub_expr3, ScalarMult(sub_expr2, TensorProd(sub_expr4, sub_expr5))))

# 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())

\left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\right) {\otimes} \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right)\right)\right) = \left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\right) {\otimes} \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right)\right)\right)

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