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, subtract, two, zero
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = frac(one, Exp(two, subtract(frac(Add(t, one), two), frac(t, two))))
sub_expr3 = VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1))
sub_expr4 = 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 = Equals(TensorProd(ScalarMult(sub_expr2, sub_expr3), TensorProd(sub_expr4)), ScalarMult(sub_expr2, TensorProd(sub_expr3, sub_expr4)))

# 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(\left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\right)\right) {\otimes} \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)\right) = \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}{\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)\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: 12 operands: 5
4	Operation	operator: 50 operands: 6
5	ExprTuple	7, 8
6	ExprTuple	14, 9
7	Operation	operator: 50 operands: 10
8	Operation	operator: 12 operands: 11
9	Operation	operator: 12 operands: 13
10	ExprTuple	14, 15
11	ExprTuple	16
12	Literal
13	ExprTuple	15, 16
14	Operation	operator: 63 operands: 17
15	Operation	operator: 37 operands: 18
16	ExprRange	lambda_map: 19 start_index: 20 end_index: 56
17	ExprTuple	71, 21
18	ExprTuple	43, 22
19	Lambda	parameter: 84 body: 23
20	Operation	operator: 59 operands: 24
21	Operation	operator: 78 operands: 25
22	Operation	operator: 50 operands: 26
23	Operation	operator: 50 operands: 27
24	ExprTuple	28, 71
25	ExprTuple	80, 29
26	ExprTuple	30, 58
27	ExprTuple	31, 32
28	Operation	operator: 82 operand: 68
29	Operation	operator: 59 operands: 34
30	Operation	operator: 78 operands: 35
31	Operation	operator: 63 operands: 36
32	Operation	operator: 37 operands: 38
33	ExprTuple	68
34	ExprTuple	39, 40
35	ExprTuple	69, 41
36	ExprTuple	71, 42
37	Literal
38	ExprTuple	43, 44
39	Operation	operator: 63 operands: 45
40	Operation	operator: 82 operand: 53
41	Operation	operator: 72 operands: 47
42	Operation	operator: 78 operands: 48
43	Operation	operator: 66 operand: 56
44	Operation	operator: 50 operands: 51
45	ExprTuple	52, 80
46	ExprTuple	53
47	ExprTuple	80, 74, 75, 77, 54
48	ExprTuple	80, 55
49	ExprTuple	56
50	Literal
51	ExprTuple	57, 58
52	Operation	operator: 59 operands: 60
53	Operation	operator: 63 operands: 61
54	Operation	operator: 78 operands: 62
55	Operation	operator: 63 operands: 64
56	Literal
57	Operation	operator: 78 operands: 65
58	Operation	operator: 66 operand: 71
59	Literal
60	ExprTuple	68, 71
61	ExprTuple	68, 80
62	ExprTuple	80, 68
63	Literal
64	ExprTuple	71, 80
65	ExprTuple	69, 70
66	Literal
67	ExprTuple	71
68	Variable
69	Literal
70	Operation	operator: 72 operands: 73
71	Literal
72	Literal
73	ExprTuple	80, 74, 75, 76, 77
74	Literal
75	Literal
76	Operation	operator: 78 operands: 79
77	Literal
78	Literal
79	ExprTuple	80, 81
80	Literal
81	Operation	operator: 82 operand: 84
82	Literal
83	ExprTuple	84
84	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 ¶