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Expression of type Equals

from the theory of proveit.physics.quantum.QPE

In [1]:
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, m
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Sum, e, frac, i, one, pi, subtract, two
from proveit.physics.quantum.QPE import _alpha_m_mod_two_pow_t, _m_domain, _phase, _two_pow_t
In [2]:
# build up the expression from sub-expressions
expr = Equals(_alpha_m_mod_two_pow_t, Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], summand = Exp(Exp(e, Mult(two, pi, i, subtract(_phase, frac(m, _two_pow_t)))), k), domain = _m_domain)))
expr:
In [3]:
# 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
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())
\alpha_{m ~\textup{mod}~ 2^{t}} = \left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} (\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\varphi - \frac{m}{2^{t}}\right)})^{k}\right)\right)
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(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')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 5
operand: 8
4Operationoperator: 33
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 10
8Operationoperator: 11
operands: 50
9Operationoperator: 49
operands: 12
10Operationoperator: 13
operand: 15
11Literal
12ExprTuple45, 52
13Literal
14ExprTuple15
15Lambdaparameter: 24
body: 17
16ExprTuple24
17Conditionalvalue: 18
condition: 19
18Operationoperator: 53
operands: 20
19Operationoperator: 21
operands: 22
20ExprTuple23, 24
21Literal
22ExprTuple24, 25
23Operationoperator: 53
operands: 26
24Variable
25Operationoperator: 27
operands: 28
26ExprTuple29, 30
27Literal
28ExprTuple31, 32
29Literal
30Operationoperator: 33
operands: 34
31Literal
32Operationoperator: 40
operands: 35
33Literal
34ExprTuple55, 36, 37, 38
35ExprTuple52, 39
36Literal
37Literal
38Operationoperator: 40
operands: 41
39Operationoperator: 46
operand: 45
40Literal
41ExprTuple43, 44
42ExprTuple45
43Literal
44Operationoperator: 46
operand: 48
45Literal
46Literal
47ExprTuple48
48Operationoperator: 49
operands: 50
49Literal
50ExprTuple51, 52
51Variable
52Operationoperator: 53
operands: 54
53Literal
54ExprTuple55, 56
55Literal
56Literal