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# Large Expressions with Greedy
Using the greedy method allows the contraction of hundreds of tensors. Here's
an example from quantum of computing the inner product between two ['Matrix
Product States'](https://en.wikipedia.org/wiki/Matrix_product_state).
Graphically, if we represent each tensor as an `O`, give it
the same number of 'legs' as it has indices, and join those legs when that
index is summed with another tensor, we get an expression for `n` particles
that looks like:
```console
O-O-O-O-O-O- -O-O-O-O-O-O
| | | | | | ... | | | | | |
O-O-O-O-O-O- -O-O-O-O-O-O
0 1 2 3 4 5 ........... n-2 n-1
```
The meaning of this is not that important other than its a large, useful
contraction. For `n=100` it involves 200 different tensors and about 300
unique indices. With this many indices it can be useful to generate them with
the function `opt_einsum.parser.get_symbol`.
### Setup the string
```python
import numpy as np
import opt_einsum as oe
n = 100
phys_dim = 3
bond_dim = 10
# start with first site
# O--
# |
# O--
einsum_str = "ab,ac,"
for i in range(1, n - 1):
# set the upper left/right, middle and lower left/right indices
# --O--
# |
# --O--
j = 3 * i
ul, ur, m, ll, lr = (oe.get_symbol(i)
for i in (j - 1, j + 2, j, j - 2, j + 1))
einsum_str += "{}{}{},{}{}{},".format(m, ul, ur, m, ll, lr)
# finish with last site
# --O
# |
# --O
i = n - 1
j = 3 * i
ul, m, ll, = (oe.get_symbol(i) for i in (j - 1, j, j - 2))
einsum_str += "{}{},{}{}".format(m, ul, m, ll)
```
### Generate the shapes
```python
def gen_shapes():
yield (phys_dim, bond_dim)
yield (phys_dim, bond_dim)
for i in range(1, n - 1):
yield(phys_dim, bond_dim, bond_dim)
yield(phys_dim, bond_dim, bond_dim)
yield (phys_dim, bond_dim)
yield (phys_dim, bond_dim)
shapes = tuple(gen_shapes())
```
Let's time how long it takes to generate the expression (`'greedy'` is used by default, and we turn off the `memory_limit`):
```python
%timeit expr = oe.contract_expression(einsum_str, *shapes, memory_limit=-1)
#> 76.2 ms ± 1.05 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
```
This is pretty manageable, though we might want to think about splitting the
expression up if we go a lot bigger.
Importantly, we can then use this repeatedly with any set of matching arrays:
```python
arrays = [np.random.randn(*shp) / 4 for shp in shapes]
expr(*arrays)
#> array(23.23628116)
arrays = [np.random.randn(*shp) / 4 for shp in shapes]
expr(*arrays)
#> array(-12.21091879)
```
### Full path
And if we **really** want we can generate the full contraction path info:
```python
print(oe.contract_path(einsum_str, *arrays, memory_limit=-1)[1])
#> Complete contraction: ab,ac,dcf,dbe,gfi,geh,jil,jhk,mlo,mkn,por,pnq,sru,sqt,vux,vtw,yxA,ywz,BAD,BzC,EDG,ECF,HGJ,HFI,KJM,KIL,NMP,NLO,QPS,QOR,TSV,TRU,WVY,WUX,ZYÂ,ZXÁ,ÃÂÅ,ÃÁÄ,ÆÅÈ,ÆÄÇ,ÉÈË,ÉÇÊ,ÌËÎ,ÌÊÍ,ÏÎÑ,ÏÍÐ,ÒÑÔ,ÒÐÓ,ÕÔ×,ÕÓÖ,Ø×Ú,ØÖÙ,ÛÚÝ,ÛÙÜ,ÞÝà,ÞÜß,áàã,áßâ,äãæ,äâå,çæé,çåè,êéì,êèë,íìï,íëî,ðïò,ðîñ,óòõ,óñô,öõø,öô÷,ùøû,ù÷ú,üûþ,üúý,ÿþā,ÿýĀ,ĂāĄ,ĂĀă,ąĄć,ąăĆ,ĈćĊ,ĈĆĉ,ċĊč,ċĉČ,ĎčĐ,ĎČď,đĐē,đďĒ,ĔēĖ,ĔĒĕ,ėĖę,ėĕĘ,ĚęĜ,ĚĘě,ĝĜğ,ĝěĞ,ĠğĢ,ĠĞġ,ģĢĥ,ģġĤ,ĦĥĨ,ĦĤħ,ĩĨī,ĩħĪ,ĬīĮ,ĬĪĭ,įĮı,įĭİ,IJıĴ,IJİij,ĵĴķ,ĵijĶ,ĸķĺ,ĸĶĹ,ĻĺĽ,ĻĹļ,ľĽŀ,ľļĿ,ŁŀŃ,ŁĿł,ńŃņ,ńłŅ,Ňņʼn,ŇŅň,ŊʼnŌ,Ŋňŋ,ōŌŏ,ōŋŎ,ŐŏŒ,ŐŎő,œŒŕ,œőŔ,ŖŕŘ,ŖŔŗ,řŘś,řŗŚ,ŜśŞ,ŜŚŝ,şŞš,şŝŠ,ŢšŤ,ŢŠţ,ťŤŧ,ťţŦ,ŨŧŪ,ŨŦũ,ūŪŭ,ūũŬ,ŮŭŰ,ŮŬů,űŰų,űůŲ,ŴųŶ,ŴŲŵ,ŷŶŹ,ŷŵŸ,źŹż,źŸŻ,Žżſ,ŽŻž,ƀſƂ,ƀžƁ,ƃƂƅ,ƃƁƄ,Ɔƅƈ,ƆƄƇ,ƉƈƋ,ƉƇƊ,ƌƋƎ,ƌƊƍ,ƏƎƑ,ƏƍƐ,ƒƑƔ,ƒƐƓ,ƕƔƗ,ƕƓƖ,ƘƗƚ,ƘƖƙ,ƛƚƝ,ƛƙƜ,ƞƝƠ,ƞƜƟ,ơƠƣ,ơƟƢ,ƤƣƦ,ƤƢƥ,ƧƦƩ,Ƨƥƨ,ƪƩƬ,ƪƨƫ,ƭƬƯ,ƭƫƮ,ưƯƲ,ưƮƱ,ƳƲƵ,ƳƱƴ,ƶƵ,ƶƴ->
#> Naive scaling: 298
#> Optimized scaling: 5
#> Naive FLOP count: 1.031e+248
#> Optimized FLOP count: 1.168e+06
#> Theoretical speedup: 88264689284468460017580864156865782413140936705854966013600065426858041248009637246968036807489558012989638169986640870276510490846199301907401763236976204166215471281505344088317454144870323271826022036197984172898402324699098341524952317952.000
#> Largest intermediate: 3.000e+02 elements
#> --------------------------------------------------------------------------------
#> scaling BLAS current remaining
#> --------------------------------------------------------------------------------
#> 4 TDOT dbe,ab->ade ac,dcf,gfi,geh,jil,jhk,mlo,mkn,por,pnq,sru,sqt,vux,vtw,yxA,ywz,BAD,BzC,EDG,ECF,HGJ,HFI,KJM,KIL,NMP,NLO,QPS,QOR,TSV,TRU,WVY,WUX,ZYÂ,ZXÁ,ÃÂÅ,ÃÁÄ,ÆÅÈ,ÆÄÇ,ÉÈË,ÉÇÊ,ÌËÎ,ÌÊÍ,ÏÎÑ,ÏÍÐ,ÒÑÔ,ÒÐÓ,ÕÔ×,ÕÓÖ,Ø×Ú,ØÖÙ,ÛÚÝ,ÛÙÜ,ÞÝà,ÞÜß,áàã,áßâ,äãæ,äâå,çæé,çåè,êéì,êèë,íìï,íëî,ðïò,ðîñ,óòõ,óñô,öõø,öô÷,ùøû,ù÷ú,üûþ,üúý,ÿþā,ÿýĀ,ĂāĄ,ĂĀă,ąĄć,ąăĆ,ĈćĊ,ĈĆĉ,ċĊč,ċĉČ,ĎčĐ,ĎČď,đĐē,đďĒ,ĔēĖ,ĔĒĕ,ėĖę,ėĕĘ,ĚęĜ,ĚĘě,ĝĜğ,ĝěĞ,ĠğĢ,ĠĞġ,ģĢĥ,ģġĤ,ĦĥĨ,ĦĤħ,ĩĨī,ĩħĪ,ĬīĮ,ĬĪĭ,įĮı,įĭİ,IJıĴ,IJİij,ĵĴķ,ĵijĶ,ĸķĺ,ĸĶĹ,ĻĺĽ,ĻĹļ,ľĽŀ,ľļĿ,ŁŀŃ,ŁĿł,ńŃņ,ńłŅ,Ňņʼn,ŇŅň,ŊʼnŌ,Ŋňŋ,ōŌŏ,ōŋŎ,ŐŏŒ,ŐŎő,œŒŕ,œőŔ,ŖŕŘ,ŖŔŗ,řŘś,řŗŚ,ŜśŞ,ŜŚŝ,şŞš,şŝŠ,ŢšŤ,ŢŠţ,ťŤŧ,ťţŦ,ŨŧŪ,ŨŦũ,ūŪŭ,ūũŬ,ŮŭŰ,ŮŬů,űŰų,űůŲ,ŴųŶ,ŴŲŵ,ŷŶŹ,ŷŵŸ,źŹż,źŸŻ,Žżſ,ŽŻž,ƀſƂ,ƀžƁ,ƃƂƅ,ƃƁƄ,Ɔƅƈ,ƆƄƇ,ƉƈƋ,ƉƇƊ,ƌƋƎ,ƌƊƍ,ƏƎƑ,ƏƍƐ,ƒƑƔ,ƒƐƓ,ƕƔƗ,ƕƓƖ,ƘƗƚ,ƘƖƙ,ƛƚƝ,ƛƙƜ,ƞƝƠ,ƞƜƟ,ơƠƣ,ơƟƢ,ƤƣƦ,ƤƢƥ,ƧƦƩ,Ƨƥƨ,ƪƩƬ,ƪƨƫ,ƭƬƯ,ƭƫƮ,ưƯƲ,ưƮƱ,ƳƲƵ,ƳƱƴ,ƶƵ,ƶƴ,ade->
#> 4 TDOT dcf,ac->adf gfi,geh,jil,jhk,mlo,mkn,por,pnq,sru,sqt,vux,vtw,yxA,ywz,BAD,BzC,EDG,ECF,HGJ,HFI,KJM,KIL,NMP,NLO,QPS,QOR,TSV,TRU,WVY,WUX,ZYÂ,ZXÁ,ÃÂÅ,ÃÁÄ,ÆÅÈ,ÆÄÇ,ÉÈË,ÉÇÊ,ÌËÎ,ÌÊÍ,ÏÎÑ,ÏÍÐ,ÒÑÔ,ÒÐÓ,ÕÔ×,ÕÓÖ,Ø×Ú,ØÖÙ,ÛÚÝ,ÛÙÜ,ÞÝà,ÞÜß,áàã,áßâ,äãæ,äâå,çæé,çåè,êéì,êèë,íìï,íëî,ðïò,ðîñ,óòõ,óñô,öõø,öô÷,ùøû,ù÷ú,üûþ,üúý,ÿþā,ÿýĀ,ĂāĄ,ĂĀă,ąĄć,ąăĆ,ĈćĊ,ĈĆĉ,ċĊč,ċĉČ,ĎčĐ,ĎČď,đĐē,đďĒ,ĔēĖ,ĔĒĕ,ėĖę,ėĕĘ,ĚęĜ,ĚĘě,ĝĜğ,ĝěĞ,ĠğĢ,ĠĞġ,ģĢĥ,ģġĤ,ĦĥĨ,ĦĤħ,ĩĨī,ĩħĪ,ĬīĮ,ĬĪĭ,įĮı,įĭİ,IJıĴ,IJİij,ĵĴķ,ĵijĶ,ĸķĺ,ĸĶĹ,ĻĺĽ,ĻĹļ,ľĽŀ,ľļĿ,ŁŀŃ,ŁĿł,ńŃņ,ńłŅ,Ňņʼn,ŇŅň,ŊʼnŌ,Ŋňŋ,ōŌŏ,ōŋŎ,ŐŏŒ,ŐŎő,œŒŕ,œőŔ,ŖŕŘ,ŖŔŗ,řŘś,řŗŚ,ŜśŞ,ŜŚŝ,şŞš,şŝŠ,ŢšŤ,ŢŠţ,ťŤŧ,ťţŦ,ŨŧŪ,ŨŦũ,ūŪŭ,ūũŬ,ŮŭŰ,ŮŬů,űŰų,űůŲ,ŴųŶ,ŴŲŵ,ŷŶŹ,ŷŵŸ,źŹż,źŸŻ,Žżſ,ŽŻž,ƀſƂ,ƀžƁ,ƃƂƅ,ƃƁƄ,Ɔƅƈ,ƆƄƇ,ƉƈƋ,ƉƇƊ,ƌƋƎ,ƌƊƍ,ƏƎƑ,ƏƍƐ,ƒƑƔ,ƒƐƓ,ƕƔƗ,ƕƓƖ,ƘƗƚ,ƘƖƙ,ƛƚƝ,ƛƙƜ,ƞƝƠ,ƞƜƟ,ơƠƣ,ơƟƢ,ƤƣƦ,ƤƢƥ,ƧƦƩ,Ƨƥƨ,ƪƩƬ,ƪƨƫ,ƭƬƯ,ƭƫƮ,ưƯƲ,ưƮƱ,ƳƲƵ,ƳƱƴ,ƶƵ,ƶƴ,ade,adf->
#> 4 GEMM ƶƵ,ƳƲƵ->ƳƶƲ gfi,geh,jil,jhk,mlo,mkn,por,pnq,sru,sqt,vux,vtw,yxA,ywz,BAD,BzC,EDG,ECF,HGJ,HFI,KJM,KIL,NMP,NLO,QPS,QOR,TSV,TRU,WVY,WUX,ZYÂ,ZXÁ,ÃÂÅ,ÃÁÄ,ÆÅÈ,ÆÄÇ,ÉÈË,ÉÇÊ,ÌËÎ,ÌÊÍ,ÏÎÑ,ÏÍÐ,ÒÑÔ,ÒÐÓ,ÕÔ×,ÕÓÖ,Ø×Ú,ØÖÙ,ÛÚÝ,ÛÙÜ,ÞÝà,ÞÜß,áàã,áßâ,äãæ,äâå,çæé,çåè,êéì,êèë,íìï,íëî,ðïò,ðîñ,óòõ,óñô,öõø,öô÷,ùøû,ù÷ú,üûþ,üúý,ÿþā,ÿýĀ,ĂāĄ,ĂĀă,ąĄć,ąăĆ,ĈćĊ,ĈĆĉ,ċĊč,ċĉČ,ĎčĐ,ĎČď,đĐē,đďĒ,ĔēĖ,ĔĒĕ,ėĖę,ėĕĘ,ĚęĜ,ĚĘě,ĝĜğ,ĝěĞ,ĠğĢ,ĠĞġ,ģĢĥ,ģġĤ,ĦĥĨ,ĦĤħ,ĩĨī,ĩħĪ,ĬīĮ,ĬĪĭ,įĮı,įĭİ,IJıĴ,IJİij,ĵĴķ,ĵijĶ,ĸķĺ,ĸĶĹ,ĻĺĽ,ĻĹļ,ľĽŀ,ľļĿ,ŁŀŃ,ŁĿł,ńŃņ,ńłŅ,Ňņʼn,ŇŅň,ŊʼnŌ,Ŋňŋ,ōŌŏ,ōŋŎ,ŐŏŒ,ŐŎő,œŒŕ,œőŔ,ŖŕŘ,ŖŔŗ,řŘś,řŗŚ,ŜśŞ,ŜŚŝ,şŞš,şŝŠ,ŢšŤ,ŢŠţ,ťŤŧ,ťţŦ,ŨŧŪ,ŨŦũ,ūŪŭ,ūũŬ,ŮŭŰ,ŮŬů,űŰų,űůŲ,ŴųŶ,ŴŲŵ,ŷŶŹ,ŷŵŸ,źŹż,źŸŻ,Žżſ,ŽŻž,ƀſƂ,ƀžƁ,ƃƂƅ,ƃƁƄ,Ɔƅƈ,ƆƄƇ,ƉƈƋ,ƉƇƊ,ƌƋƎ,ƌƊƍ,ƏƎƑ,ƏƍƐ,ƒƑƔ,ƒƐƓ,ƕƔƗ,ƕƓƖ,ƘƗƚ,ƘƖƙ,ƛƚƝ,ƛƙƜ,ƞƝƠ,ƞƜƟ,ơƠƣ,ơƟƢ,ƤƣƦ,ƤƢƥ,ƧƦƩ,Ƨƥƨ,ƪƩƬ,ƪƨƫ,ƭƬƯ,ƭƫƮ,ưƯƲ,ưƮƱ,ƳƱƴ,ƶƴ,ade,adf,ƳƶƲ->
#> 4 GEMM ƶƴ,ƳƱƴ->ƳƶƱ gfi,geh,jil,jhk,mlo,mkn,por,pnq,sru,sqt,vux,vtw,yxA,ywz,BAD,BzC,EDG,ECF,HGJ,HFI,KJM,KIL,NMP,NLO,QPS,QOR,TSV,TRU,WVY,WUX,ZYÂ,ZXÁ,ÃÂÅ,ÃÁÄ,ÆÅÈ,ÆÄÇ,ÉÈË,ÉÇÊ,ÌËÎ,ÌÊÍ,ÏÎÑ,ÏÍÐ,ÒÑÔ,ÒÐÓ,ÕÔ×,ÕÓÖ,Ø×Ú,ØÖÙ,ÛÚÝ,ÛÙÜ,ÞÝà,ÞÜß,áàã,áßâ,äãæ,äâå,çæé,çåè,êéì,êèë,íìï,íëî,ðïò,ðîñ,óòõ,óñô,öõø,öô÷,ùøû,ù÷ú,üûþ,üúý,ÿþā,ÿýĀ,ĂāĄ,ĂĀă,ąĄć,ąăĆ,ĈćĊ,ĈĆĉ,ċĊč,ċĉČ,ĎčĐ,ĎČď,đĐē,đďĒ,ĔēĖ,ĔĒĕ,ėĖę,ėĕĘ,ĚęĜ,ĚĘě,ĝĜğ,ĝěĞ,ĠğĢ,ĠĞġ,ģĢĥ,ģġĤ,ĦĥĨ,ĦĤħ,ĩĨī,ĩħĪ,ĬīĮ,ĬĪĭ,įĮı,įĭİ,IJıĴ,IJİij,ĵĴķ,ĵijĶ,ĸķĺ,ĸĶĹ,ĻĺĽ,ĻĹļ,ľĽŀ,ľļĿ,ŁŀŃ,ŁĿł,ńŃņ,ńłŅ,Ňņʼn,ŇŅň,ŊʼnŌ,Ŋňŋ,ōŌŏ,ōŋŎ,ŐŏŒ,ŐŎő,œŒŕ,œőŔ,ŖŕŘ,ŖŔŗ,řŘś,řŗŚ,ŜśŞ,ŜŚŝ,şŞš,şŝŠ,ŢšŤ,ŢŠţ,ťŤŧ,ťţŦ,ŨŧŪ,ŨŦũ,ūŪŭ,ūũŬ,ŮŭŰ,ŮŬů,űŰų,űůŲ,ŴųŶ,ŴŲŵ,ŷŶŹ,ŷŵŸ,źŹż,źŸŻ,Žżſ,ŽŻž,ƀſƂ,ƀžƁ,ƃƂƅ,ƃƁƄ,Ɔƅƈ,ƆƄƇ,ƉƈƋ,ƉƇƊ,ƌƋƎ,ƌƊƍ,ƏƎƑ,ƏƍƐ,ƒƑƔ,ƒƐƓ,ƕƔƗ,ƕƓƖ,ƘƗƚ,ƘƖƙ,ƛƚƝ,ƛƙƜ,ƞƝƠ,ƞƜƟ,ơƠƣ,ơƟƢ,ƤƣƦ,ƤƢƥ,ƧƦƩ,Ƨƥƨ,ƪƩƬ,ƪƨƫ,ƭƬƯ,ƭƫƮ,ưƯƲ,ưƮƱ,ade,adf,ƳƶƲ,ƳƶƱ->
#> 5 TDOT ade,geh->adgh gfi,jil,jhk,mlo,mkn,por,pnq,sru,sqt,vux,vtw,yxA,ywz,BAD,BzC,EDG,ECF,HGJ,HFI,KJM,KIL,NMP,NLO,QPS,QOR,TSV,TRU,WVY,WUX,ZYÂ,ZXÁ,ÃÂÅ,ÃÁÄ,ÆÅÈ,ÆÄÇ,ÉÈË,ÉÇÊ,ÌËÎ,ÌÊÍ,ÏÎÑ,ÏÍÐ,ÒÑÔ,ÒÐÓ,ÕÔ×,ÕÓÖ,Ø×Ú,ØÖÙ,ÛÚÝ,ÛÙÜ,ÞÝà,ÞÜß,áàã,áßâ,äãæ,äâå,çæé,çåè,êéì,êèë,íìï,íëî,ðïò,ðîñ,óòõ,óñô,öõø,öô÷,ùøû,ù÷ú,üûþ,üúý,ÿþā,ÿýĀ,ĂāĄ,ĂĀă,ąĄć,ąăĆ,ĈćĊ,ĈĆĉ,ċĊč,ċĉČ,ĎčĐ,ĎČď,đĐē,đďĒ,ĔēĖ,ĔĒĕ,ėĖę,ėĕĘ,ĚęĜ,ĚĘě,ĝĜğ,ĝěĞ,ĠğĢ,ĠĞġ,ģĢĥ,ģġĤ,ĦĥĨ,ĦĤħ,ĩĨī,ĩħĪ,ĬīĮ,ĬĪĭ,įĮı,įĭİ,IJıĴ,IJİij,ĵĴķ,ĵijĶ,ĸķĺ,ĸĶĹ,ĻĺĽ,ĻĹļ,ľĽŀ,ľļĿ,ŁŀŃ,ŁĿł,ńŃņ,ńłŅ,Ňņʼn,ŇŅň,ŊʼnŌ,Ŋňŋ,ōŌŏ,ōŋŎ,ŐŏŒ,ŐŎő,œŒŕ,œőŔ,ŖŕŘ,ŖŔŗ,řŘś,řŗŚ,ŜśŞ,ŜŚŝ,şŞš,şŝŠ,ŢšŤ,ŢŠţ,ťŤŧ,ťţŦ,ŨŧŪ,ŨŦũ,ūŪŭ,ūũŬ,ŮŭŰ,ŮŬů,űŰų,űůŲ,ŴųŶ,ŴŲŵ,ŷŶŹ,ŷŵŸ,źŹż,źŸŻ,Žżſ,ŽŻž,ƀſƂ,ƀžƁ,ƃƂƅ,ƃƁƄ,Ɔƅƈ,ƆƄƇ,ƉƈƋ,ƉƇƊ,ƌƋƎ,ƌƊƍ,ƏƎƑ,ƏƍƐ,ƒƑƔ,ƒƐƓ,ƕƔƗ,ƕƓƖ,ƘƗƚ,ƘƖƙ,ƛƚƝ,ƛƙƜ,ƞƝƠ,ƞƜƟ,ơƠƣ,ơƟƢ,ƤƣƦ,ƤƢƥ,ƧƦƩ,Ƨƥƨ,ƪƩƬ,ƪƨƫ,ƭƬƯ,ƭƫƮ,ưƯƲ,ưƮƱ,adf,ƳƶƲ,ƳƶƱ,adgh->
#>
#> ...
#>
#> 4 TDOT Ğğ,ĠğĢ->ĠĞĢ ĠĞġ,ģĢĥ,ģġĤ,Ĥĥ,ĠĞĢ->
#> 4 GEMM ĠĞĢ,ĠĞġ->ġĢ ģĢĥ,ģġĤ,Ĥĥ,ġĢ->
#> 4 GEMM Ĥĥ,ģĢĥ->ģĢĤ ģġĤ,ġĢ,ģĢĤ->
#> 4 TDOT ģĢĤ,ģġĤ->ġĢ ġĢ,ġĢ->
#> 2 DOT ġĢ,ġĢ-> ->
```
Where we can see the speedup over a naive einsum is about `10^241`, not bad!
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