Local Algorithms for 1D Quantum Systems
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1 Local Algorithms for 1D Quantum Systems I. Arad, Z. Landau, U. Vazirani, T. Vidick I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 1 / 14
2 Many-body physics Are physical systems special? All states Physically relevant states I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 2 / 14
3 Ground States of Local Hamiltonians Hilbert space of particles H = (C d ) n, Local terms 0 H i 1, Interested in lowest eignevectors of H = P i Hi. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 3 / 14
4 Ground States of Local Hamiltonians Hilbert space of particles H = (C d ) n, Local terms 0 H i 1, Interested in lowest eignevectors of H = P i Hi. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 3 / 14
5 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
6 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
7 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. QMA-complete for 1D (gap 1 poly(n) )[ 07] I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
8 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. QMA-complete for 1D (gap Area law for 1D with unique ground state and constant gap[ 07]. 1 poly(n) )[ 07] I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
9 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. QMA-complete for 1D (gap Area law for 1D with unique ground state and constant gap[ 07]. Polynomial time algorithm for finding unique ground state with constant gap[ 13]. 1 poly(n) )[ 07] I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
10 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. QMA-complete for 1D (gap Area law for 1D with unique ground state and constant gap[ 07]. Polynomial time algorithm for finding unique ground state with constant gap[ 13]. What happens for critical systems (gap 0)? 1 poly(n) )[ 07] I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
11 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. QMA-complete for 1D (gap Area law for 1D with unique ground state and constant gap[ 07]. Polynomial time algorithm for finding unique ground state with constant gap[ 13]. What happens for critical systems (gap 0)? Phase transition: believed "Log correction to area law" 1 poly(n) )[ 07] I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
12 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. QMA-complete for 1D (gap Area law for 1D with unique ground state and constant gap[ 07]. Polynomial time algorithm for finding unique ground state with constant gap[ 13]. What happens for critical systems (gap 0)? Phase transition: believed "Log correction to area law" QMA-complete for 1D (gap 1 poly(n) )[ 07] 1 poly(n) ) I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
13 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. QMA-complete for 1D (gap Area law for 1D with unique ground state and constant gap[ 07]. Polynomial time algorithm for finding unique ground state with constant gap[ 13]. What happens for critical systems (gap 0)? Phase transition: believed "Log correction to area law" Density of states assumptions area laws QMA-complete for 1D (gap 1 poly(n) )[ 07] 1 poly(n) ) I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
14 1D Hamiltonians Second eigenstate Ground State constant Second eigenstate Ground State 1 poly(n) DMRG does well in practice [ 92]. QMA-complete for 1D (gap Area law for 1D with unique ground state and constant gap[ 07]. Polynomial time algorithm for finding unique ground state with constant gap[ 13]. What happens for critical systems (gap 0)? Phase transition: believed "Log correction to area law" Density of states assumptions area laws QMA-complete for 1D (gap 1 poly(n) )[ 07] 1 poly(n) ) not too many states Ground State I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 4 / 14
15 Result: Algorithm for critical systems poly(n) eigenvalues in constant interval poly(n) states below some constant energy Ground State I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 5 / 14
16 Result: Algorithm for critical systems poly(n) eigenvalues in constant interval poly(n) states below some constant energy Ground State 2 poly(log n) time algorithm for finding low energy states, poly(log n) MPS description with bond dimension 2 I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 5 / 14
17 Result: Algorithm for critical systems poly(n) eigenvalues in constant interval poly(n) states below some constant energy Ground State 2 poly(log n) time algorithm for finding low energy states, poly(log n) MPS description with bond dimension 2 Notable features: More physically plausible: No convex optimization. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 5 / 14
18 Result: Algorithm for critical systems poly(n) eigenvalues in constant interval poly(n) states below some constant energy Ground State 2 poly(log n) time algorithm for finding low energy states, poly(log n) MPS description with bond dimension 2 Notable features: More physically plausible: No convex optimization. Hierarchical structure reminiscent of Renormalization Group. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 5 / 14
19 Result: Algorithm for degenerate ground space Degenerate ground space with gap: I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 6 / 14
20 Result: Algorithm for degenerate ground space Degenerate ground space with gap: Generalization of unique ground state case? I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 6 / 14
21 Result: Algorithm for degenerate ground space Degenerate ground space with gap: Generalization of unique ground state case? Small perturbation would have 1/poly(n) gap I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 6 / 14
22 Result: Algorithm for degenerate ground space Degenerate ground space with gap: Generalization of unique ground state case? Small perturbation would have 1/poly(n) gap Area law and algorithm for constant degeneracy [Huang; Chubb, Flammia]. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 6 / 14
23 Result: Algorithm for degenerate ground space Degenerate ground space with gap: Generalization of unique ground state case? Small perturbation would have 1/poly(n) gap Area law and algorithm for constant degeneracy [Huang; Chubb, Flammia]. poly(n) degenerate ground space next eigenstate constant poly(n) dimensional ground space I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 6 / 14
24 Result: Algorithm for degenerate ground space Degenerate ground space with gap: Generalization of unique ground state case? Small perturbation would have 1/poly(n) gap Area law and algorithm for constant degeneracy [Huang; Chubb, Flammia]. poly(n) degenerate ground space next eigenstate constant poly(n) dimensional ground space Polynomial time algorithm for finding the ground space. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 6 / 14
25 Key tool: Approximate Ground State Projection (AGSP) I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 7 / 14
26 Key tool: Approximate Ground State Projection (AGSP) It approximately projects onto the ground state: I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 7 / 14
27 Key tool: Approximate Ground State Projection (AGSP) It approximately projects onto the ground state: Ground State Ground state Perpendicular Space Eigenvalues of AGSP 1 Eigenvalues of H I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 7 / 14
28 Key tool: Approximate Ground State Projection (AGSP) It approximately projects onto the ground state: Ground State Ground state 1 Eigenvalues of AGSP Perpendicular Space Eigenvalues of H It has small entanglement rank: D K I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 7 / 14
29 Key tool: Approximate Ground State Projection (AGSP) It approximately projects onto the ground state: Ground State Ground state 1 Eigenvalues of AGSP Perpendicular Space Eigenvalues of H It has small entanglement rank: D K Critical threshold D < 1. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 7 / 14
30 Steps towards an algorithm A first pass. 1 Start with unentangled state. 2 Apply AGSP. Randomly pick one Schmidt vector. Repeat. D K I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 8 / 14
31 Steps towards an algorithm A first pass. 1 Start with unentangled state. 2 Apply AGSP. Randomly pick one Schmidt vector. Repeat. D K Each iteration moves closer Area Law. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 8 / 14
32 Steps towards an algorithm A first pass. 1 Start with unentangled state. 2 Apply AGSP. Randomly pick one Schmidt vector. Repeat. D K Each iteration moves closer Area Law. Algorithm? I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 8 / 14
33 Steps towards an algorithm A first pass. 1 Start with unentangled state. 2 Apply AGSP. Randomly pick one Schmidt vector. Repeat. D K Each iteration moves closer Area Law. Algorithm? Will take O(n) iterations to get close. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 8 / 14
34 Steps towards an algorithm A first pass. 1 Start with unentangled state. 2 Apply AGSP. Randomly pick one Schmidt vector. Repeat. D K Each iteration moves closer Area Law. Algorithm? Will take O(n) iterations to get close. Complexity OK at the cut but too complex on sides. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 8 / 14
35 Steps towards an algorithm A first pass. 1 Start with unentangled state. 2 Apply AGSP. Randomly pick one Schmidt vector. Repeat. D K Each iteration moves closer Area Law. Algorithm? Will take O(n) iterations to get close. Complexity OK at the cut but too complex on sides. Two directions: I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 8 / 14
36 Steps towards an algorithm A first pass. 1 Start with unentangled state. 2 Apply AGSP. Randomly pick one Schmidt vector. Repeat. D K Each iteration moves closer Area Law. Algorithm? Will take O(n) iterations to get close. Complexity OK at the cut but too complex on sides. Two directions: 1 Reduce complexity original algorithm. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 8 / 14
37 Steps towards an algorithm A first pass. 1 Start with unentangled state. 2 Apply AGSP. Randomly pick one Schmidt vector. Repeat. D K Each iteration moves closer Area Law. Algorithm? Will take O(n) iterations to get close. Complexity OK at the cut but too complex on sides. Two directions: 1 Reduce complexity original algorithm. 2 Reduce iterations new results. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 8 / 14
38 Original algorithm: bird s eye view I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 9 / 14
39 Original algorithm: bird s eye view S 1 Viable set: subspace containing left Schmidt vectors of a good approximation to ground space I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 9 / 14
40 Original algorithm: bird s eye view S 2 Viable set: subspace containing left Schmidt vectors of a good approximation to ground space I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 9 / 14
41 Original algorithm: bird s eye view S i Viable set: subspace containing left Schmidt vectors of a good approximation to ground space I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 9 / 14
42 Original algorithm: bird s eye view Viable set: subspace containing left Schmidt vectors of a good approximation to ground space S n I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 9 / 14
43 Reduce iterations Move more quickly via simultaneous local movement? I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 10 / 14
44 Reduce iterations Move more quickly via simultaneous local movement? Tree Structure: I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 10 / 14
45 Reduce iterations Move more quickly via simultaneous local movement? Tree Structure: Need: a way to merge two viable sets. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 10 / 14
46 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 11 / 14
47 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces Merge Process I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 11 / 14
48 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces Merge Process 1 Merge: Tensor two neighboring viable sets V = V 1 V 2, I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 11 / 14
49 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces Merge Process 1 Merge: Tensor two neighboring viable sets V = V 1 V 2, 2 Size Reduction: I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 11 / 14
50 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces Merge Process 1 Merge: Tensor two neighboring viable sets V = V 1 V 2, 2 Size Reduction: 1 Choose a small random subspace of V V, I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 11 / 14
51 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces Merge Process 1 Merge: Tensor two neighboring viable sets V = V 1 V 2, 2 Size Reduction: 1 Choose a small random subspace of V V, 2 "Apply AGSP" write K = P D i,j=1 A i B i,j C j, and choose V = span{b i,j V }. Viable Set AGSP I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 11 / 14
52 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 12 / 14
53 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces Modified AGSP AGSP Need poly(n) bond dimension inside I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 12 / 14
54 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces Modified AGSP AGSP Requires soft truncation Need poly(n) bond dimension inside Uses [Molnar, Schuch, Verstraete, Cirac]: e βh has small bond dimension MPO approximation. I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 12 / 14
55 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 13 / 14
56 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces V a viable set for subspace T For every element t T : a vector with left-schmidt vectors in V that is near t and projects exactly in the direction t. t Ground Space I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 13 / 14
57 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces V a viable set for subspace T For every element t T : a vector with left-schmidt vectors in V that is near t and projects exactly in the direction t. Viable Set t Ground Space I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 13 / 14
58 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces V a viable set for subspace T For every element t T : a vector with left-schmidt vectors in V that is near t and projects exactly in the direction t. Viable Set t Ground Space I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 13 / 14
59 New Algorithm Ingredients Merge process Modified AGSP Viable set for subspaces V a viable set for subspace T For every element t T : a vector with left-schmidt vectors in V that is near t and projects exactly in the direction t. Viable Set t Ground Space I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 13 / 14
60 Discussion next eigenstate poly(n) states below some constant energy constant Ground State poly(n) dimensional ground space Physics viewpoint: when do density of states conditions occur? Examples when they don t? Proofs? Could this be a skeleton for a practical algorithm? Are there interesting questions in this regime to answer? What kinds of challenges exist? What kinds of questions/connections does the tree-like structure of the algorithm suggest? I. Arad, Z. Landau, U. Vazirani, T. Vidick () Local Algorithms for 1D Quantum Systems 14 / 14
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