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Computing Numerical and Exact SIC-POVMs

Chaos and Quantum Information Seminar Jagiellonian University Krak´ow

Computing Numerical and Exact SIC-POVMs Markus Grassl International Centre for Theory of Quantum Technologies University Gdansk [email protected] sicpovm.markus-grassl.de

29 March 2021

in collaboration with Marcus Appleby, Ingemar Bengtsson, Michael Harrison, Gary McConnell, Andrew Scott additional support by the Max Planck Institute for the Science of Light, Erlangen, and MPG

29.03.2021 – 1– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs)

General measurement device (from a quantum information processing point of view)

single-shot experiment

✲ “1”

✲ “2” one “click” ✲ ✲ ρin “3” per input ✲ “4”

✲ “5”

29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs)

General measurement device (from a quantum information processing point of view)

single-shot experiment probabilities ✲ “1” p1 Tr(ρinΠ1) ✲  “2” p2 Tr(ρinΠ2)  one “click”  ρ ✲ ✲  in “3” p3 = Tr(ρinΠ3) 0  = Πi 0 per input ≥  ⇒ ≥ ✲ “4” p4 Tr(ρinΠ4)  ✲  “5” p5 Tr(ρinΠ5)   

29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs)

General measurement device (from a quantum information processing point of view)

single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) ✲  “2” p2 = Tr(ρinΠ2)  one “click”  ρ ✲ ✲  in “3” p3 = Tr(ρinΠ3) 0  = Πi 0 per input ≥  ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4)  ✲  “5” p5 = Tr(ρinΠ5)   

29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs)

General measurement device (from a quantum information processing point of view)

single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) 0 ≥ ✲  “2” p2 = Tr(ρinΠ2) 0 ≥  one “click”  ρ ✲ ✲  in “3” p3 = Tr(ρinΠ3) 0  = Πi 0 per input ≥  ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) 0 ≥  ✲  “5” p5 = Tr(ρinΠ5) 0  ≥  

29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs)

General measurement device (from a quantum information processing point of view)

single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) 0 ≥ ✲  “2” p2 = Tr(ρinΠ2) 0 ≥  one “click”  ρ ✲ ✲  in “3” p3 = Tr(ρinΠ3) 0  = Πi 0 per input ≥  ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) 0 ≥  ✲  “5” p5 = Tr(ρinΠ5) 0  ≥  

29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs)

General measurement device (from a quantum information processing point of view)

single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) 0 ≥ ✲  “2” p2 = Tr(ρinΠ2) 0 ≥  one “click”  ρ ✲ ✲  in “3” p3 = Tr(ρinΠ3) 0  = Πi 0 per input ≥  ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) 0 ≥  ✲  “5” p5 = Tr(ρinΠ5) 0  ≥   1= pi = Tr(ρin Πi) P P

29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs)

General measurement device (from a quantum information processing point of view)

single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) 0 ≥ ✲  “2” p2 = Tr(ρinΠ2) 0 ≥  one “click”  ρ ✲ ✲  in “3” p3 = Tr(ρinΠ3) 0  = Πi 0 per input ≥  ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) 0 ≥  ✲  “5” p5 = Tr(ρinΠ5) 0  ≥   1= pi = Tr(ρ Πi) = Πi = 1 in ⇒ P P P

29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs)

m • set of m positive semideﬁnite operators Πi with Πi = 1 iP=1 • implementation with the help of an m-dimensional auxiliary system and a projective (von Neumann) measurement on the auxiliary system

† † ρ 0 0 U (ρ 0 0 ) U Aiρ A i i in ⊗| ih | −→ in ⊗| ih | −→ in i ⊗| ih | † with Ai Ai = Πi (not unique), depending on U Neumark extension • note: the number m of outcomes is independent of the dimension d of the

Hilbert space of ρin

29.03.2021 – 3– Markus Grassl Computing Numerical and Exact SIC-POVMs (Minimal) Informationally-Complete POVMs

2 • POVM with m = d linearly independent operators Πi = unique representation of any state as ⇒ d2

ρ = cjΠj Xj=1

• measuring the POVM yields probabilities

d2 pi = Tr(ρ Πi)= cj Tr(ΠiΠj ) Xj=1

= reconstructing ρ by inverting the matrix with elements Tr(ΠiΠj) ⇒

29.03.2021 – 4– Markus Grassl Computing Numerical and Exact SIC-POVMs Symmetric Informationally-Complete POVMs

• informationally complete POVM with rank-one elements Πi = α ψi ψi | ih | • symmetry: overlaps tr(ΠiΠj )= const = β for i = j 6 • 1 1 using Πi = 1 it follows that α = d and β = d2(d+1) P 2 1+ δijd = ψi ψj = ⇒ |h | i| 1+ d • reconstruction

d2 ρ = d(d + 1) Tr(ρ Πj) 1 Πj − Xj=1 =cj | {z } = coeﬃcient cj is an aﬃne function of the probability pj = tr(ρ Πj) ⇒

29.03.2021 – 5– Markus Grassl Computing Numerical and Exact SIC-POVMs A Simple to State Problem

2 d Are there d vectors v , v ,..., v 2 C in the complex vector space of 1 2 d ∈ dimension d such that:

2 (i) vj vj = 1 for j = 1,...,d h | i 2 1 2 (ii) vj vk = for 1 j

The vectors vj form an equiangular tight frame/ﬁnite unit norm tight frame.

29.03.2021 – 6– Markus Grassl Computing Numerical and Exact SIC-POVMs A Simple to State Problem

2 d Are there d vectors v , v ,..., v 2 C in the complex vector space of 1 2 d ∈ dimension d such that:

2 (i) vj vj = 1 for j = 1,...,d h | i 2 1 2 (ii) vj vk = for 1 j

The vectors vj form an equiangular tight frame/ﬁnite unit norm tight frame.

All solutions form a real algebraic variety, using 2d real variables per vector

T vj =(aj + ibj ,aj + ibj ,...,aj,d + ibj,d) (i = √ 1) 1 1 2 2 − 2 3 2 d 2d variables, d equations (i) of degree 2 and 2 equations (ii) of degree 4.

29.03.2021 – 6– Markus Grassl Computing Numerical and Exact SIC-POVMs Weyl-Heisenberg Group

• generators: Hd := X,Z h i

d−1 d−1 where X := j + 1 j and Z := ωj j j | ih | d| ih | jP=0 jP=0 (ωd := exp(2πi/d)) • relations:

′ ′ c a b c′ a′ b′ a b−b a c′ a′ b′ c a b ωdX Z ωd X Z = ωd ωd X Z ωdX Z • basis:

a b Hd ζ(Hd)= X Z : a,b 0,...,d 1 = Zd Zd ∈ { − } ∼ × trace-orthogonal basis of all d d matrices ×

29.03.2021 – 7– Markus Grassl Computing Numerical and Exact SIC-POVMs Constructing SIC-POVMs

Ansatz:

SIC-POVM that is the orbit under the Weyl-Heisenberg group Hd, i.e.,

v(a,b) := XaZb v(0,0) | i | i ′ ′ ′ ′ 1 for (a,b)=(a ,b ), v(a,b) v(a ,b ) 2 =  |h | i| 1/(d + 1) for (a,b) =(a′,b′) 6

d−1 (0,0) v = (x j + ix j ) j , | i 2 2 +1 | i Xj=0

(x0,...,x2d−1 are real variables, x1 = 0)

= we have to ﬁnd only one ﬁducial vector v(0,0) instead of d2 vectors ⇒ | i = polynomial equations with 2d 1 variables, but already quite complicated ⇒ − for d = 6

29.03.2021 – 8– Markus Grassl Computing Numerical and Exact SIC-POVMs Jacobi Group (or Cliﬀord Group)

• automorphism group of the Weyl-Heisenberg group Hd, i.e.

† T Jd : T HdT = Hd ∀ ∈

• the action of Jd on Hd modulo phases corresponds to the symplectic group SL(2, Zd), i.e.

′ ′ ′ a a † a b c a b ˜ ˜ T X Z T = ωdX Z where   = T  , T SL(2, Zd) b′ b ∈     = homomorphism Jd SL(2, Zd) ⇒ → • additionally: complex conjugation (anti-unitary)

1 0 XaZb XaZ−b corresponding to   7→ 0 1  − 

29.03.2021 – 9– Markus Grassl Computing Numerical and Exact SIC-POVMs Zauner’s Conjecture

[G. Zauner, Dissertation, Universit¨at Wien, 1999] Conjecture: For every dimension d 2 there exists a SIC-POVM whose elements are the ≥ orbit of a rank-one operator E0 under the Weyl-Heisenberg group Hd.

What is more, E0 commutes with an element S of the Jacobi group Jd. The action of S on Hd modulo the center has order three.

support for this conjecture (to date): • numerical solutions for all dimensions d 193, plus a few more ≤ • exact algebraic solutions for some dimensions (see below)

one of the prize problems in [Pawel Horodecki,Lukasz Rudnicki, Karol Zyczkowski,˙ Five open problems in quantum information, arXiv:2002.03233]

29.03.2021 – 10– Markus Grassl Computing Numerical and Exact SIC-POVMs Numerical Search for SIC-POVMs

• for any state ψ Cd | i∈ d d 2 f( ψ )= ψ j + ℓ ℓ ψ ψ k + ℓ j + k + ℓ ψ | i h | ih | ih | ih | i j,kX=1 Xℓ=1 d d 2 2 = ψj+ℓ ψℓ ψk+ℓ ψj+k+ℓ ≥ d + 1 j,kX=1 Xℓ=1

with equality iﬀ ψ is a ﬁducial vector for a Weyl-Heisenberg SIC-POVM | i • gradient descent to minimize f( ψ ), subject to unit norm | i

29.03.2021 – 11– Markus Grassl Computing Numerical and Exact SIC-POVMs Numerical Search for SIC-POVMs

• for any state ψ Cd | i∈ d d 2 2 f( ψ )= ψj+ℓ ψℓ ψk+ℓ ψj+k+ℓ | i ≥ d + 1 j,kX=1 Xℓ=1

with equality iﬀ ψ is a ﬁducial vector for a Weyl-Heisenberg SIC-POVM | i • gradient descent to minimize f( ψ ), subject to unit norm | i P ~x • use F (~x)= f for an arbitrary vector ~x Cd, P ~x ∈ where P is the projectionk k onto a subspace (prescribed symmetry) • chain rule yields a relatively simple formula for the gradient of F (~x) in terms of the gradient of f • complexity O(d3) for both the function and the gradient when storing O(d2) intermediate values

29.03.2021 – 11– Markus Grassl Computing Numerical and Exact SIC-POVMs Numerical Search for SIC-POVMs

• eﬃcient implementation of F (~x) and its gradient in C++ by Andrew Scott • parallel computation of the function/gradient using OpenMP/CUDA • minimization using limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm • search runs into local minima, we need many random initial points • running many instances on HPC clusters by MPG and GWDG • – for d = 189: approx. 23.3 106 trials, 3.48 CPU years × – for d = 190: approx. 66.8 106 trials, 10.51 CPU years × – for d = 193: approx. 78.3 106 trials, 13.00 CPU years × – for d = 5779: 55065 trials, 17.69 GPU years, no success

29.03.2021 – 12– Markus Grassl Computing Numerical and Exact SIC-POVMs Average Number of Iterations

average number of iterations (100 solutions) 1×109 avg. no sym 1.22**x 1×108

1×107

1×106

100000

iterations 10000

1000

100

10

1 0 10 20 30 40 50 60 70 80 90 100 dimension

29.03.2021 – 13– Markus Grassl Computing Numerical and Exact SIC-POVMs Average Number of Iterations

average number of iterations (100 solutions) 1×109 avg. no sym 1.22**x 1×108 avg. Zauner 1.09**x

1×107

1×106

100000

iterations 10000

1000

100

10

1 0 10 20 30 40 50 60 70 80 90 100 dimension

29.03.2021 – 14– Markus Grassl Computing Numerical and Exact SIC-POVMs Average Number of Iterations with Zauner Symmetry

number of iterations with Zauner symmetry (100 solutions) 4500 avg max 4000 min

3500

3000

2500

2000 iterations 1500

1000

500

0

−500 0 10 20 30 40 50 60 70 dimension

29.03.2021 – 15– Markus Grassl Computing Numerical and Exact SIC-POVMs Average Number of Iterations with Zauner Symmetry

number of iterations with Zauner symmetry (100 solutions) 40000 avg max 35000 min

30000

25000

20000

iterations 15000

10000

5000

0

−5000 50 55 60 65 70 75 80 85 90 95 dimension

29.03.2021 – 16– Markus Grassl Computing Numerical and Exact SIC-POVMs Fibonacci-Lucas SIC-POVMs

[Markus Grassl & Andrew J. Scott, JMP 58, December 2017, arXiv:1707.02944]

• (exact) symmetry analysis of a numerical solution for d = 124 = symmetry group of order 30 (prescribed order 6) ⇒ • identiﬁed as part of a series of dimensions (related to Lucas numbers) d = 4, 8, 19, 48, 124, 323, 844, 2208, 5779, 15128, 39604,... • symmetry group of order 6k related to Fibonacci numbers

• new exact solutions for d = 124 and d = 323 (previously d = 48) • new numerical solutions for d = 844 and d = 2208 (previously d = 323)

29.03.2021 – 17– Markus Grassl Computing Numerical and Exact SIC-POVMs Fibonacci-Lucas SIC-POVMs

[Markus Grassl & Andrew J. Scott, JMP 58, December 2017, arXiv:1707.02944]

• Fibonacci numbers Fk with F0 = 0, F1 = 1, Fk+1 = Fk + Fk−1

• Lucas numbers Lk with L0 = 2, L1 = 1, Lk+1 = Lk + Lk−1 • prescribed anti-unitary symmetry related to the Fibonacci matrix

0 1 Fk−1 Fk A =   Ak =   1 1 Fk Fk+1    

• modulo dk = L2k + 1, the matrix A has order 6k • sequence of dimensions d = 4, 8, 19, 48, 124, 323, 844, 2208, 5779, 15128, 39604,... • squarefree part D of (d + 1)(d 3) is always D = 5 − = ray class ﬁeld over Q(√5) ⇒

29.03.2021 – 18– Markus Grassl Computing Numerical and Exact SIC-POVMs Generalised Fibonacci-Lucas SIC-POVMs

• generalised Fibonacci numbers Fm,k with

Fm,0 = 0, Fm,1 = 1, Fm,k+1 = mFm,k + Fm,k−1

• generalised Lucas numbers Lm,k with

Lm,0 = 2, Lm,1 = m, Lm,k+1 = mLm,k + Lm,k−1 • prescribed anti-unitary symmetry related to the matrix

0 1 k Fm,k−1 Fm,k Am =   Am =   1 m Fm,k Fm,k+1    

• modulo dm,k = Lm,2k + 1, the matrix Am has order 6k • squarefree part D of (d + 1)(d 3) equals the squarefree part of m2 + 4 − = ray class ﬁeld over Q(√D) ⇒

29.03.2021 – 19– Markus Grassl Computing Numerical and Exact SIC-POVMs Dimensions with Anti-Unitary Symmetries

k 123 4 56 7 8 ord(F ) 6 12 18 24 30 36 42 48

m D Fe′ Fg 1 5 4 8 19 48 124 323 844 2208 2 2 7 35 199 1155 6727 39203 228487 1331715 3 13 12 120 1299 14160 154452 1684803 18378372 200477280 4 5 19 323 5779 103683 1860499 5 29 28 728 19603 528528 14250628 6 10 39 1443 54759 2079363 78960999 7 53 52 2600 132499 6754800 8 17 67 4355 287299 18957315 9 85 84 6888 571539 47430768 10 26 103 10403 1060903 11 5 124 15128 1860499 12 37 147 21315 3111699 13 173 172 29240 4999699 14 2 199 39203 7761799 15 229 228 51528 16 65 259 66563 17 293 292 84680 18 82 327 106275 19 365 364 131768 20 101 403 161603

29.03.2021 – 20– Markus Grassl Computing Numerical and Exact SIC-POVMs Families of SIC-POVMs with Unitary Symmetry

• prescribed unitary symmetry related to the matrix

0 1 Bm =   1 m −  • k similar recurrence relations for the entries of Bm and the corresponding dimension • order of the symmetry is 3k

29.03.2021 – 21– Markus Grassl Computing Numerical and Exact SIC-POVMs Dimensions with Unitary Symmetries

k 123456 7 8 9 ord(F ) 3 6 9 12 15 18 21 24 27

m D Fz Fb Fd 3 5 4 8 19 48 124 323 844 2208 5779 4 3 5 15 53 195 725 2703 10085 37635 140453 5 21 6 24 111 528 2526 12099 57966 277728 1330671 6 2 7 35 199 1155 6727 39203 228487 1331715 7761799 7 5 8 48 323 2208 15128 103683 710648 4870848 33385283 8 15 9 63 489 3843 30249 238143 1874889 14760963 116212809 9 77 10 80 703 6240 55450 492803 4379770 38925120 345946303 10 6 11 99 971 9603 95051 940899 9313931 92198403 912670091 11 13 12 120 1299 14160 154452 12 35 13 143 1693 20163 240253 13 165 14 168 2159 27888 360374 14 3 15 195 2703 37635 524175 15 221 16 224 3331 49728 742576 16 7 17 255 4049 64515 17 285 18 288 4863 82368 18 5 19 323 5779 103683 19 357 20 360 6803 128880 20 11 21 399 7941 158403

29.03.2021 – 22– Markus Grassl Computing Numerical and Exact SIC-POVMs Symmetries and Ray Class Fields

[Appleby, Chien, Flammia & Waldron, J. Phys. A. 51, 2018, arXiv:1703.05981] Ray class field conjecture nested tower of fields (for the minimal field)

′ Q ✁ IEc = Q(√D) ✁ IE0 ✁ IE1 ✁ IE= IE1(i√d ).

• IE is the ray class field over Q(√D) with conductor d′ with ramification at both infinite places, a2D =(d + 1)(d 3) − • IE1 is the ray class field with ramification only allowed at the infinite place taking √D to a positive real number

• IE0 is the Hilbert class ﬁeld over Q(√D), in particular [IE0 : Q(√D)] equals the class number of Q(√D)

29.03.2021 – 23– Markus Grassl Computing Numerical and Exact SIC-POVMs Symmetries and Ray Class Fields

[Appleby, Chien, Flammia & Waldron, J. Phys. A. 51, 2018, arXiv:1703.05981] Ray class field conjecture nested tower of fields (for the minimal field)

′ Q ✁ IEc = Q(√D) ✁ IE0 ✁ IE1 ✁ IE= IE1(i√d ).

• for a certain maximal Abelian subgroup of GL(2, Z/d′Z) and M (essentially) the symmetry group S(Π) of the SIC-POVM:

Gal(IE /IE ) = /S(Π) 1 0 ∼ M our observation: estimate for the group order:

Gal(IE /IEc) S(Π) = |M| = |M|×| 0 | | | Gal(IE /IE ) Gal(IE /IEc) | 1 0 | | 1 | • 1 or 4 cases for , but S(Π) must be integral |M| | |

29.03.2021 – 24– Markus Grassl Computing Numerical and Exact SIC-POVMs Symmetries of SIC-POVMs 51 estimate 2208 additional ✻ 48 anti-unitary symmetry unitary 45 ``double'' anti-unitary 844 42 39 323 1299 36 33 124 30 27 48 120 1155 24 21 19 199 2703 18 725 15 8 12 35 195 12 53 9 4 7 15 6 5 3 0 ✲ 10 100 1000 dimension

29.03.2021 – 25– Markus Grassl Computing Numerical and Exact SIC-POVMs Symmetries of SIC-POVMs 75 72 estimate additional ✻ anti-unitary symmetry 69 unitary 39604 66 ``double'' anti-unitary 63 15128 60 57 5779 54 51 2208 48 45 844 42 39 323 36 33 124 30 27 48 24 21 19 18 15 8 12 9 4 6 3 0 10 100 1000 10000 100000✲ dimension

29.03.2021 – 26– Markus Grassl Computing Numerical and Exact SIC-POVMs Known SIC-POVMs 51 solutions 2208 additional ✻ 48 dimension/50 symmetry dimension/60 45 dimension/70 844 42 39 323 36 33 124 30 27 48 1155 24 21 19 199 1299 18 15 8 35 120 12 9 4 7 12 6 3 0 ✲ 1 10 100 1000 dimension

29.03.2021 – 27– Markus Grassl Computing Numerical and Exact SIC-POVMs Exact Solutions from Numerical Solutions

[Appleby, Chien, Flammia & Waldron, J. Phys. A. 51, 2018, arXiv:1703.05981]

• ray class ﬁeld conjecture: Gal(IE /IE ) = /S(Π) 1 0 ∼ M ′ • for σ Gal(IE /IE ), we have a matrix Gσ GL(2, Z/d Z) such that ∈ 1 0 ∈

σ (Tr(ΠDp)) = Tr(ΠDGσ p)

• expansion coefficients cp = Tr(ΠDp) in the same orbit under are M related by Galois conjugation • the coefficients of the polynomial fp0 (z)= (z cp) M − p∈Qp0 lie in a number field of “small” degree • find the exact minimal polynomials of those coefficients (requires high-precision numerical solution) • find the roots of the exact polynomials fp0 (z) in the ray class field

29.03.2021 – 28– Markus Grassl Computing Numerical and Exact SIC-POVMs More Exact Solutions from Numerical Solutions [Markus Grassl, arXiv:210?.????]

• when Gσ has determinant 1: † † σ (Tr(ΠDp)) = Tr(ΠDGσ p) = Tr(ΠTσDpTσ) = Tr(TσΠTσDp) = action of T † on the projection Π and the state ψ ⇒ σ | i • when Tσ is a permutation matrix, we obtain a Galois action on the cofficients of the state ψ | i = similar approach for polynomials with the coefficients of ψ as roots ⇒ | i • new exact solutions for 57 additional dimensions (so far) d = 26, 38, 42, 49, 52, 57, 61, 62, 63, 65, 67, 73, 74, 78, 79, 84, 86, 91, 93, 95, 97, 98, 103, 109, 111, 122, 127, 129, 133, 134, 139, 143, 146, 147, 151, 155, 157, 163, 168, 169, 172, 181, 182, 183, 193, 199, 201, 228, 259, 292, 327, 364, 399, 403, 489, 844, 1299 • fiducial vectors lie in a proper (“small”) subfield of the ray class field

29.03.2021 – 29– Markus Grassl Computing Numerical and Exact SIC-POVMs Direct Conversion of the Fiducial Vector

[M. Appleby & I. Bengtsson, Simpliﬁed exact SICs, JMP 60, June 2019, arXiv:1811.00947]

• start with a numerical ﬁducial vector • instead of the standard basis (of the Weyl-Heisenberg group), express the vector with respect to an adapted basis of the eigenspace of the symmetry • for some dimensions, the absolute values and phases/ratios of phases of the expansion coeﬃcients have “nice” exact minimal polynomials • demonstrated for d = 5, 15, 195 (with D = 3) by Appleby & Bengtsson

• additional exact solution for d = 725 (also with D = 3), final result in number field of degree 134 400 • difficulties – additional square roots are required for intermediate results – so far unclear for which dimensions this will work

29.03.2021 – 30– Markus Grassl Computing Numerical and Exact SIC-POVMs Hot Oﬀ the Fire

new recipes under development with Marcus Appleby, Ingemar Bengtsson, Michael Harrison, Gary McConnell

• analysing exact solutions in certain dimensions led to new conjectures relating SIC-POVMs and number theory • does not require a numerical ﬁducial vector • avoids factoring exact polynomials over number ﬁelds of large degree • intermediate steps require (so far) high precision • new solutions: – d = 5579 (2021-02-12) – d = 787 (2021-02-12) – d = 487 (2021-02-18)

29.03.2021 – 31– Markus Grassl Computing Numerical and Exact SIC-POVMs Current State of the SIC-POVM Problem

• numerical solutions: – d 45: Joesph Renes et al. quant-ph/0310075 ≤ – d 67: Andrew Scott & Markus Grassl arXiv:0910.5784 ≤ – d 121 plus a few more: Andrew Scott arXiv:1703.03993 ≤ – d 151: Christopher Fuchs et al. arXiv:1703.07901 ≤ – d 193, d = 204, 224, 255, 288, 528, 1155, 2208 ≤ • exact algebraic solutions: – early 2017: d = 2–24, 28, 30, 31, 35, 37, 39, 43, 48, 124 (32 dimensions) – all dimensions d 53, 57, 61, 62, 63, 65, 67, 73, 74, 76, 78, 79, 80, ≤ 84, 86, 91, 93, 95, 97, 98, 99, 103, 109, 111, 120, 122, 124, 127, 129, 133, 134, 139, 143, 146, 147, 151, 155, 157, 163, 168, 169, 172, 181, 182, 183, 193, 195, 199, 201, 228, 259, 292, 323, 327, 364, 399, 403, 487 , 489, 725, 787 , 844, 1299, 5799 (115 dimensions)

29.03.2021 – 32– Markus Grassl Computing Numerical and Exact SIC-POVMs Thank you! Danke! Merci! Dziekuje! Acknowledgments The ‘International Centre for Theory of Quantum Technologies’ project (contract no. 2018/MAB/5) is carried out within the International Research Agendas Programme of the Foundation for Polish Science co-ﬁnanced by the European Union from the funds of the Smart Growth Operational Programme, axis IV: Increasing the research potential (Measure 4.3).

Additional computing resources were provided via MPL

29.03.2021 – 33– Markus Grassl

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