https://www.snowdropquantum.com/tangledblog daily 0.75 2026-02-01 https://www.snowdropquantum.com/tangledblog/the-alphaq-up-challenge monthly 0.5 2026-02-03 https://www.snowdropquantum.com/tangledblog/every-new-beginning-comes-from-some-other-beginnings-end monthly 0.5 2026-01-17 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/ca025dc3-7356-431c-9554-6f601d92a71f/20250801_132206%280%29.jpg TANGLED - Every New Beginning Comes From Some Other Beginning's End - Make it stand out Me during the part of the walk where I went through Vancouver. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/510797cc-ccec-4786-bce4-ca441a329a00/petersen.png TANGLED - Every New Beginning Comes From Some Other Beginning's End - Make it stand out Look at how cool it is!!!! https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/9dc51f84-3ad1-4374-b35b-e2e57f7fdfe3/graph_5_head_to_head.png TANGLED - Every New Beginning Comes From Some Other Beginning's End - Make it stand out Figure 1: Full round-robin tournament results for all nine agents. The best agent, AlphaZero_100 GT, lost 24 games playing as red and 8 games playing as blue. Interestingly it did not lose any games to any of the other AlphaZero agents. This result probably means it just hasn’t been trained long enough to see the strategies the other agents are using, and if I kept training it, it would probably at some point become unbeatable. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/9c0100b0-c0ea-4278-bd60-cc0ff72a9ac8/graph_5_final_tournament_rankings.png TANGLED - Every New Beginning Comes From Some Other Beginning's End - Make it stand out Figure 2. Ranked agents from the tournament. https://www.snowdropquantum.com/tangledblog/fun-with-spotify-and-math monthly 0.5 2025-05-08 https://www.snowdropquantum.com/tangledblog/pushing-into-uncharted-territory monthly 0.5 2025-05-05 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e2079ec9-5653-4529-be13-3f0f6f25ccca/Screenshot+2025-04-24+195857.png TANGLED - Pushing Into Uncharted Territory - Make it stand out Cube fixed vertex Tangled game graph. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/3413e1ed-ccc6-4a2f-a6aa-4e9bc1a391b1/Screenshot+2025-05-04+104703.png TANGLED - Pushing Into Uncharted Territory - Make it stand out Results of the round-robin tournament, showing each agent’s results against each other agent both playing as red and blue. The way to read this is that the blue-colored cells represent W/L/D for the blue-colored agent, and the red-colored cells represent W/L/D for the red-colored agent. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/181748b2-0018-4c2f-94f8-5da60162a0d4/Screenshot+2025-05-04+213038.png TANGLED - Pushing Into Uncharted Territory - Make it stand out Summary of results, ranked in descending Elo order. The number in the agent name is the number of online rollouts used per action selection. The advantage for red is interesting as it implies going first provides an advantage. The top four agents are essentially equivalent to each other, producing mostly draws (although the ranking seems solid in that a higher ranked agent reliably does better, even marginally, than those immediately below it). https://www.snowdropquantum.com/tangledblog/first-quantum-agent-deployed monthly 0.5 2025-04-12 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/4faf8107-4264-4e2f-bbf8-7c92cdaf0a21/Screenshot+2025-03-28+085717.png TANGLED - My First Quantum Agent - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/0e6b774d-b3c4-4793-83ea-34d727cb831d/bloodsport.jpg TANGLED - My First Quantum Agent - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/b8aa6b38-f118-4d92-87e3-0fb52ac7024c/Screenshot+2025-04-07+083631.png TANGLED - My First Quantum Agent - Make it stand out Results of the round-robin tournament, showing each agent’s results against each other agent both playing as red and blue. The way to read this is that the blue-colored cells represent W/L/D for the blue-colored agent, and the red-colored cells represent W/L/D for the red-colored agent. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/db083b24-6841-46e3-91ff-935b8b7e4613/Screenshot+2025-04-07+132738.png TANGLED - My First Quantum Agent - Make it stand out Summary of results, ranked in descending Elo order. The number in the agent name is the number of online rollouts used per action selection. The key result here is that agents trained on and using quantum annealing (QA) results perform significantly better than their counterparts using simulated annealing (SA). Also, there’s a huge advantage to playing red on this game graph. https://www.snowdropquantum.com/tangledblog/analysis-of-adjudication-of-terminal-states-for-small-graphs monthly 0.5 2025-04-05 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d604d617-0917-4c01-9a63-9c880d82305d/graph_11_RSG.png TANGLED - Analysis of Adjudication of Terminal States for Small Graphs - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/0de5bef6-5b51-46ea-a8a1-6f26eb888c83/graph_2_K3.png TANGLED - Analysis of Adjudication of Terminal States for Small Graphs - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/027f764f-bf17-4e70-81b3-3309d1fd15bc/graph_20_Glued_Triangles.png TANGLED - Analysis of Adjudication of Terminal States for Small Graphs - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/de28c6af-8327-4ce5-86de-02cf5432f494/graph_3_K4.png TANGLED - Analysis of Adjudication of Terminal States for Small Graphs - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/86e9ee7e-05e6-4865-ab2b-38ca14f0ce58/graph_3_K4_close.png TANGLED - Analysis of Adjudication of Terminal States for Small Graphs - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d1b18ce5-7516-48ee-9c99-c6073097984c/graph_19_barbell.png TANGLED - Analysis of Adjudication of Terminal States for Small Graphs - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e4c13d33-6783-4d72-8961-f2cb91ee39dd/graph_19_barbell_close.png TANGLED - Analysis of Adjudication of Terminal States for Small Graphs - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/modifying-tangled-with-fixed-token-positions monthly 0.5 2025-04-05 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d35aaa4e-7f31-4358-9c24-17c1adc1b236/Screenshot+2025-03-28+091504.png TANGLED - Modified Tangled With Fixed Token Positions - Make it stand out Here are my choices for red/blue vertices for the ‘fixed vertex variant’ of Tangled. The graph number is my internal label for the graphs — they don’t make sense except they are numbered in the order I added them. The names of the graphs are a combination of usual names and ones I made up. The (V, E) is the number of vertices and edges in each graph. Anything bigger than the cube graph is too big to enumerate all the terminal states. These bigger graphs are important but at this stage I’m focusing on the smaller ones just so I understand what’s going on. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/5958f415-4f7e-45cf-84ab-c5c6b4e51241/Screenshot+2025-03-28+095317.png TANGLED - Modified Tangled With Fixed Token Positions - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/5b0c8ed1-150e-41e9-8f97-0276a0398b40/Screenshot+2025-03-29+213205.png TANGLED - Modified Tangled With Fixed Token Positions - Make it stand out For the Moser Spindle: MCTS1000 red and MCTS10 blue: 485 / 0 / 11 and MCTS 1000 blue and MCTS10 red: 441 / 0 / 55 — a small advantage to playing red (makes sense as red has one more move (6 vs 5) for this graph). For the Cube Graph where each player takes 6 moves: MCTS1000 red and MCTS10 blue: 439 / 0 / 57 and MCTS1000 blue and MCTS10 red: 443 / 0 / 53 — looks like no advantage to either. https://www.snowdropquantum.com/tangledblog/there-is-no-special-role-for-observers-in-quantum-mechanics monthly 0.5 2025-03-29 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/50eefbc4-4a3a-4e98-89b4-043a6464d419/Screenshot+2025-02-17+092310.png TANGLED - Some Thoughts on Quantum Mechanics and Consciousness - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/setting-up-elo-scores-for-a-new-game monthly 0.5 2025-02-18 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/0a17b4c7-1972-4a28-aab3-5c95f9ba4707/ArpadElo.jpg TANGLED - Setting up Elo Scores For a New Game - This is Arpad Elo. Apparently his own Elo rating peaked around 2100-2200, making him very good but not grandmaster (2500+) level. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/4c44a63d-8b08-46df-a946-07f274194a66/Screenshot+2025-02-09+132139.png TANGLED - Setting up Elo Scores For a New Game - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/59e96d74-3093-494c-801e-3706c9cbd490/rating-curve.gif TANGLED - Setting up Elo Scores For a New Game - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/873af516-e91a-4396-88e5-b4aa67be0742/equation+%287%29.png TANGLED - Setting up Elo Scores For a New Game - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e63d77c6-f398-4d35-94e1-caf25e2b9ddb/Screenshot+2025-02-18+073339.png TANGLED - Setting up Elo Scores For a New Game - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/snarks monthly 0.5 2025-02-14 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1739466278979-N3XRP4YM8C1Z8M0DIEXI/graph_5.jpg TANGLED - Snarks! - Make it stand out The smallest snark — the 10 vertex 15 edge Petersen graph. You can try coloring each edge one of three colors (for example, red, green, blue) and what you will find is that you can’t do that without two edges of the same color being connected to a vertex. Also you can’t rearrange the position of the vertices to remove all crossing edges. Interestingly those two things that sound very different are actually two different aspects of the same thing! https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d5e43f2f-beb2-4b4a-a19a-f41e852b7d42/Baphosimb.svg.png TANGLED - Snarks! - Make it stand out The sigil of Baphomet, which is not exactly the Petersen Graph, but is fairly close. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1739466336067-KOLJA6Y3WMG6Q58GGG7C/graph_13.jpg TANGLED - Snarks! - Make it stand out The 18 vertex 27 edge second Blanusa snark. Yes, there is a first Blanusa snark also. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1739466353958-7NR0RXHZPC0H2Z7IUSCG/graph_14.jpg TANGLED - Snarks! - Make it stand out The 22 vertex 33 edge first Loupekine snark. And again yes, there is a second one. Also yes, it does look like a codpiece, and if you want to get me one with this snark on it, I will wear it. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1739466379868-V3G8CJNWOBBV657XUZKO/graph_15.jpg TANGLED - Snarks! - Make it stand out The 50 vertex 75 edge Szekeres snark. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1739466398462-P174UQS1BGYAW28AUQLP/graph_16.jpg TANGLED - Snarks! - Make it stand out The 210 vertex 315 edge Descartes snark. https://www.snowdropquantum.com/tangledblog/how-i-decide-what-to-build monthly 0.5 2025-02-06 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/0e7d71e1-55bf-4492-bfc1-6f1600a0f8e5/232290711CThreadripper7000Series_4K.jpg TANGLED - Deconstructing AlphaZero Part 5: The Joys and Horrors of Parallelization - Make it stand out Threadripper Pro 7995WX, 96 cores, 5.3GHz clock, 384MB of L3 cache, and 128 PCIe Gen 5 lanes. Want. https://www.snowdropquantum.com/tangledblog/deconstructing-alphazero-part-4-how-an-agent-learns monthly 0.5 2025-01-13 https://www.snowdropquantum.com/tangledblog/building-a-game-playing-ai-for-tangled-rzgtp-x69jy-dzjkd monthly 0.5 2025-01-07 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/fad75f57-3712-4b91-9b45-5cc341cc2056/Screenshot+2024-12-21+151122.png TANGLED - Deconstructing AlphaZero Part 3: How You Gonna Act Like That - Make it stand out The neural net part of AlphaZero takes the current state s (for Tangled on the RSG the initial game state S₀ = [0,0,0, 0,0,0, 0,0,0]), feeds it into an input layer (the yellow box), and outputs prior probabilities P(S₀, A₀) (green box) and value V(S₀) (single blue neuron). The specifics of the neural net that connects input to outputs are up to you. For 2D boards where shapes are important, like Go, using convolutional layers makes sense. For graph inputs like in Tangled, graph neural networks could make sense. For now, just think of this as a feedforward system — I’ll describe the loss function and training process later when we discuss the learning process. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/32fb67bd-7ed0-4618-833f-0965bdac1308/equation+%286%29.png TANGLED - Deconstructing AlphaZero Part 3: How You Gonna Act Like That - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/58b997ab-1bcd-4e1f-8b97-59c4930582bc/equation+%283%29.png TANGLED - Deconstructing AlphaZero Part 3: How You Gonna Act Like That - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/58b997ab-1bcd-4e1f-8b97-59c4930582bc/equation+%283%29.png TANGLED - Deconstructing AlphaZero Part 3: How You Gonna Act Like That - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e4a24660-1b4e-4d7a-ba4f-3ecb5472b933/equation+%284%29.png TANGLED - Deconstructing AlphaZero Part 3: How You Gonna Act Like That - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/da1d72d6-ab66-468e-b3a4-55fd0e3d742c/equation+%285%29.png TANGLED - Deconstructing AlphaZero Part 3: How You Gonna Act Like That - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/building-a-game-playing-ai-for-tangled-rzgtp-x69jy monthly 0.5 2024-12-29 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1735415687667-1UYHBUPX5OLQQ1FBBB0P/intro_to_rl.png TANGLED - Deconstructing AlphaZero Part 2: State, Actions, and Reward, and Dreaming of Game Trees - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/bc47b55c-7d82-4a23-8c01-5219dfe5db75/Screenshot+2024-12-17+093401.png TANGLED - Deconstructing AlphaZero Part 2: State, Actions, and Reward, and Dreaming of Game Trees - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/f3395209-03d5-434f-8bd1-bff3bf2feaf6/Screenshot+2024-12-28+161832.png TANGLED - Deconstructing AlphaZero Part 2: State, Actions, and Reward, and Dreaming of Game Trees - Make it stand out This is an example of how I am writing down state for Tangled, specifically for the RSG (all other graphs are similar, just with more numbers). Just take the current game board, which has numbered vertices. For each vertex, write a +1 into state if it’s owned by red, a -1 if it’s owned by blue, and a 0 if it’s unclaimed. Then follow this by looking at each edge in the graph in lexical order and writing down three bits per edge, where the three bits are 0,0,0 if the edge is uncolored, 0,0,1 if the edge is colored gray, 0,1,0 if the edge is colored green, and 0,0,1 if the edge is colored purple. In the above example on the RSG, we start with edge (0,1) and since it’s unselected (that’s what the dashed line is supposed to mean) we write three zeros into state, and then the last edge (1,2) is purple so we write 0,0,1 into state. For the RSG, since there are always three vertices and two edges, state can always be written down as nine numbers. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/bc47b55c-7d82-4a23-8c01-5219dfe5db75/Screenshot+2024-12-17+093401.png TANGLED - Deconstructing AlphaZero Part 2: State, Actions, and Reward, and Dreaming of Game Trees - All games start with an empty board; the initial state is always [0,0,0, 0,0,0, 0,0,0] — three vertices unselected, two edges unselected. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/3f67140a-26a5-4bab-b9af-bb28e78fcdc3/Screenshot+2024-12-21+133042.png TANGLED - Deconstructing AlphaZero Part 2: State, Actions, and Reward, and Dreaming of Game Trees - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/254dcbd9-d412-4988-b51f-12caa2c32557/Screenshot+2024-12-25+073932.png TANGLED - Deconstructing AlphaZero Part 2: State, Actions, and Reward, and Dreaming of Game Trees - Make it stand out The entire Tangled game tree, including only unique states (mirror symmetrical states not shown) in lexical order (there are 1+5+21+33+27=87 unique states). Whose move it is is shown using the colored arrows. Who wins each terminal state is shown on the right. https://www.snowdropquantum.com/tangledblog/building-a-game-playing-ai-for-tangled-rzgtp monthly 0.5 2024-12-28 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/266d7d3f-4115-443e-b4e4-ec4bd3794df9/intro_to_rl.png TANGLED - Deconstructing AlphaZero Part 1: Introduction - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/520fbf1d-4af6-4cca-bde2-2a281b9f1fd5/connect4.png TANGLED - Deconstructing AlphaZero Part 1: Introduction - This is an example of a state in the game of 7x6 Connect 4. If a ‘time tick’ is a move, and the empty board is S₀, this is a specific legal arrangement of the pieces on the game board after 16 ticks (so a possible example of S₁₆). There are 4,531,985,219,092 legal 7x6 Connect 4 states, but each specific one is simple and easy to write down. If our agent is asked to make a move, just knowing the current state is enough to know everything it needs to know! https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/c12be8a4-02ae-4b31-a321-5209b07bdcb5/044A_KRAMER_BOB_DYLAN_CE_03129_l.jpg TANGLED - Deconstructing AlphaZero Part 1: Introduction - How does it feel… To be in this state https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/44d4f008-45d6-4c31-8b71-8ccd05bee203/Screenshot+2024-12-27+143545.png TANGLED - Deconstructing AlphaZero Part 1: Introduction - Make it stand out Figure 1 from Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm. The 700,000 steps represent 44, 24, and 21 million self-played games for Chess, Shogi, and Go respectively, and amounts to on the order of a few days of training for each game to go from scratch to superhuman mastery of each game. https://www.snowdropquantum.com/tangledblog/a-fun-math-problem monthly 0.5 2024-12-20 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e83fe88f-e1e0-415e-ae42-6b003ddaa9a7/Figure_1.jpg TANGLED - A Fun Math Problem - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/54ae3547-b018-4624-b1cf-85e2be0602e2/Screenshot+2024-12-19+092622.png TANGLED - A Fun Math Problem - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/62bcfcf9-ba1e-4cf6-88ba-0a2e1a8c1f2c/Screenshot+2024-12-19+094015.png TANGLED - A Fun Math Problem - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/69c90619-7320-40f3-9fe7-a6435fae9aa4/Screenshot+2024-12-19+100452.png TANGLED - A Fun Math Problem - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e4e3bf04-68c5-4cf8-ba3b-c265a064e4a0/Screenshot+2024-12-19+102601.png TANGLED - A Fun Math Problem - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/5543805a-1284-498d-9220-2011bf7be8f5/Screenshot+2024-12-19+104150.png TANGLED - A Fun Math Problem - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/10b77a90-e80c-4200-b7c1-6c878d21fb64/equation.png TANGLED - A Fun Math Problem - Make it stand out Burnside’s lemma applied to this problem. Here |G| is the number of symmetries, Fix is the number of objects that remain the same under the symmetry, I is the identity symmetry, and swap is the symmetry under swapping the two black vertices. https://www.snowdropquantum.com/tangledblog/evaluating-tangled-terminal-states-with-simulated-annealing-and-a-d-wave-quantum-computer monthly 0.5 2024-11-19 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/ae694e1d-e780-4eeb-9515-4406c9bfb8e5/equation+%281%29.png TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e232c393-e704-4c85-a4f8-cfb9d5dc4907/Graph_2_All_Unique_States.jpg TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - Make it stand out I ran all three solver methods to adjudicate all 27 unique terminal states for the three-vertex graph. This is a stacked histogram with the results using 200 bins. Also included are green dotted lines denoting the border between wins and draws (anything in between the green lines is a draw). You can see that all solvers agree on win/loss/draw for all terminal states. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/da291e40-639a-4d5e-b7a1-ed43df2f9d0a/Screenshot+2024-09-27+120749.png TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - Make it stand out I worked out the energies, correlations and influence scores for the top graph (state [0, 1, 2, 2, 3, 2]) (bottom one is similar but correlation between 0 and 1 is negative, correlation between 0 and 2 and 1 and 2 are positive, and influence is zero except for site 2 where it's positive). https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e944b55f-ca4c-4907-af8d-043bc067574c/Graph_3_All_Unique_States.jpg TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - Make it stand out Some interesting stuff here! Like what’s up with the big simulated annealing spikes near ±4/3 without accompanying quantum samples? And also why are the cyan samples not stacked on the ±2/3 result? What’s with the clumps of cyan around ±5/2? https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/dce6ac63-47a4-4696-a8b5-284c2bc646c1/Screenshot+2024-11-18+103842.jpg TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - Make it stand out This is one of the states in the red spike at score ~ -4/3. The instance has six ground states, resulting in the simulated annealer equally sampling from all six, producing an influence vector of (1,1,-1/3,1) and a score of -4/3. But this is not what the Schrodinger solver returns. Instead, in the adiabatic (long anneal time) limit, the highlighted states (1,1,-1,1) and (-1,-1,1,-1) are twice as likely as the other four, giving a different influence vector of (1,1,-1,1), and a score of -2! https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1d1e242c-34e2-407c-a32a-337b6521dab4/Screenshot+2024-11-18+105015.png TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - Note that the highlighted states are 1 spin flip away from two different states, whereas the non-highlighted ones are 1 spin flip away from only one. The transverse term in the Hamiltonian splits these six states, putting more weight into states that have more 1-spin-flip connections to other states in the groundstate bundle. This is a quantum mechanical effect, and predicts the probabilities to be 1/8 for the unfavored states and 1/4 for the favored ones in the adiabatic limit, which is what I observe with the Schrodinger solver. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d7fa7718-fc1b-4432-a58b-9ca94da1d246/se_and_sa_scores.png TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - The Schrodinger Equation results (red dots) show the score monotonically decreasing towards the adiabatic value. The orange line is just the simulated annealing result (~-4/3). https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/03b01690-8857-4353-a182-bd409abbf48e/qc_scores.png TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - This is the hardware result. It looks nothing like the Schrodinger Equation result! I think that (a) the hardware can’t sweep fast enough to randomize the states (score=0) and (b) the effect that leads to the previous graph is extremely fragile to noise on the transverse fields themselves — this different kind of noise would wash out the fragile cancellations required to get the Schrodinger result; and (c) in the long anneal time limit, both quantum and thermal effects will compete so I’d expect what we see here — something between -2 and -4/3. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/4073c3b2-b255-417f-9ccf-50da72ab2eea/Graph_3_All_Unique_States_M_10.jpg TANGLED - Adjudicating Tangled Terminal States on Tiny Graphs With Three Different Approaches - Make it stand out Computing scores over all unique terminal states for the 4-vertex complete graph, but this time using M=10 and N=1,000, no gauge transform and no shimming. Looks like nothing much changed with more samples! https://www.snowdropquantum.com/tangledblog/how-i-solve-problems-using-d-wave-hardware monthly 0.5 2024-11-16 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/0735f992-1830-4507-bf2d-db951e74dd4d/Screenshot+2024-10-18+064832.jpg TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - In what follows here, I’ll use the very simple three-vertex source graph shown here. But the process I’m describing applies to any source graph in the same way. In Tangled, this graph represents the game state [0,0,0,1,2,3] (no vertices chosen by either player; for the couplers, grey=1 is zero coupling, green=2 is ferromagnetic, and purple=3 is antiferromagnetic). https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/a434f8b7-8954-42fd-816b-9c8e2ba2ca5b/Screenshot+2024-11-12+151550.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out This depicts a randomly chosen automorphism of the problem instance we are solving (in this case the vertex permutation {1:0, 0:1, 2:2} embedded P times into hardware. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/5a71a456-dacb-4995-b2d8-bec5a47f067b/equation+%2838%29.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d988d7b1-acff-487c-a3b9-623c747ee8af/Screenshot+2024-11-12+154557.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out For our V=3 source graph, there are 2³=8 spin reversal gauges, shown here for the automorphism {1:0, 0:1, 2:2}. Black circles mean we ‘flipped’ what +1 and -1 mean for that vertex, and to preserve the symmetry we have to flip the coupler colors for all couplers connected to that vertex. You can see for this problem instance, there are only four different settings of couplers that are send to hardware, but in general this grows exponentially with coupler count. All of these are mathematically the same problem instance. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/817e1760-ac37-4130-8ac5-b73e3f1dd91c/Screenshot+2024-11-12+155038.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out After we’ve randomly ‘flipped’ each qubit with 50% probability and changed the J values appropriately, we have the situation in the cyan box where P copies of the original automorphism of the problem instance are each turned into one of the possible gauge transformed equivalent versions. Remember that each of these copies is the same problem instance mathematically, but physically they are different because of noise and analog stuff going on in the chip. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/315fb5b2-a57c-4275-9b1d-429b5105988a/mean_abs_val_of_qubit_mags.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out This is an example of the effect of shimming on the magnetizations of the qubits. This was obtained during a run on Instance 2 (defined below). Orange shows ten different values of the sum over the absolute values of all the qubit magnetizations normalized to 1 over ten runs without shimming, and the blue shows the same quantity but for the last 10 runs after shimming for S=40 runs. There’s a clear positive effect in balancing the qubits from doing the shim. Shimming unfortunately is computationally costly, roughly increasing the wallclock time by a factor proportional to S. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/3ac613bc-2e5f-4cc1-96cf-2a27c38ac832/Screenshot+2024-11-12+180607.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Instance 1 has two ground states ((1,1,1) and (-1,-1,-1)). Instance 2 has six ground states (all possible states except (1,1,-1) and (-1,-1,1)). For both, player 1 (red) selects vertex 1 and player 2 (blue) selects vertex 2. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/5492159b-5b2c-4b54-87ad-aee93a46b044/equation.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - For instance 1, the correlation between any two spins is always 1 because they are all locked in the same direction. The influence vector is (2,2,2), and the score is 2-2 = 0. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/19753549-553c-42de-9fbb-d753c8de0e8e/equation+%281%29.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - For instance 2, it’s a little harder to see, but if you assume equal occupation of all six ground states this is what the correlation matrix looks like. The influence vector is (0,0,2/3) and the score is 0 - 2/3 = -2/3. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1e97cf14-889b-4a63-a973-36103d4a71a9/instance_1.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out Scores for 20 runs for each image / choice of parameters for Instance 1. The green dashed line shows the exact score (0). The obvious weird thing here is that with no shimming and no gauge transforms the scores are all perfect, whereas for all the others there are small deviations (note the score x-axis is fairly blown up here). I’m not sure why this is… but it seems to be a real effect. One potential cause is that the instances with gauge transforms introduce anti-ferromagnetic couplers (recall Instance 1 has all ferromagnetic couplers), which could be weaker in practice than the ferromagnetic ones by a tiny bit. Also, the shimming process could potentially bias qubits away from balancing a small number of times (I’m only doing 10 shim steps so this could happen). But it is a bit disturbing, might have to keep this in mind later on if something weird happens. It looks like all the score distributions are balanced around the exact score. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/815257e0-3b18-4b74-bee4-83aec9f6fa30/Instance_2.png TANGLED - How I Adjudicate Tangled Terminal States Using D-Wave Hardware - Make it stand out Same as the previous image, but for Instance 2 where the exact score is -2/3. The width of the score distributions is about an order of magnitude larger, and the weird perfection for no shim no gauge transform goes away (this is consistent with the anti-ferromagnetic couplers potentially being worse than the ferromagnetic ones; in this Instance there are anti-ferromagnetic couplers around all the time). While the sample size here isn’t huge, eyeballing this it looks like none of my fancy stuff helped much in this case. The naive use of the hardware performs the same as all of the others. For these very small instances that get embedded in many places in the chip, the noise cancellation from just the embedding is probably doing most of the work, and the other stuff doesn’t add much. This could change for larger instances! Also again just eyeballing it, but each of these distributions looks balanced around the exact score (averaging over all 20 runs looks very close to the exact score for all of these parameter choices). It could be that the spread in runs is mostly due to the extremely low values of N I used here. https://www.snowdropquantum.com/tangledblog/some-very-useful-tools monthly 0.5 2024-09-27 https://www.snowdropquantum.com/tangledblog/is-it-possible-to-build-a-one-person-billion-dollar-company monthly 0.5 2024-09-27 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/642d26e3-3620-4797-bf7c-da5e5971c5c7/Designer+%2825%29.jpeg TANGLED - Is it Possible to Build a One-Person Billion Dollar Company? - Make it stand out This is exactly how I picture this would look. https://www.snowdropquantum.com/tangledblog/embedding-multiple-small-problems-onto-a-d-wave-processor monthly 0.5 2024-09-27 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/972127cc-74ff-4241-a876-ed4a240777a6/Screenshot+2024-09-26+162328.png TANGLED - Parallelizing Computation Using D-Wave Processors Via Multiple Simultaneous Embeddings - Make it stand out On the right in the background is the hardware graph G in grey (it's big and complicated!) where vertices 301, 306 and 307 form a triangle with the same pattern as the graph H we started with (with vertices 0, 1, and 2). The map of H into G is called an embedding into the hardware. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/f6ad3751-4ae1-403a-93e2-0aa5e81eb252/Screenshot+2024-09-26+162843.png TANGLED - Parallelizing Computation Using D-Wave Processors Via Multiple Simultaneous Embeddings - Make it stand out In addition to the original embedding, we add one to hardware vertices 591, 596, and 597. Now we have two copies of the original problem in the hardware at the same time, so one run gives us two answers! https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/96a27f9e-570f-4cad-b837-d80989399068/Screenshot+2024-09-26+163231.png TANGLED - Parallelizing Computation Using D-Wave Processors Via Multiple Simultaneous Embeddings - Make it stand out 346 embeddings of the three-vertex H into the D-Wave Advantage2.4 processor. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d48a577a-8534-49de-82a9-c3c375318a22/Screenshot+2024-09-26+164641.png TANGLED - Parallelizing Computation Using D-Wave Processors Via Multiple Simultaneous Embeddings - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/the-computational-cost-of-evaluating-terminal-states-in-tangled monthly 0.5 2024-09-27 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/6fbdfa43-3514-4f0b-a88a-0db704b39646/Screenshot+2024-06-15+221939.png TANGLED - The Computational Cost of Using the Schrodinger Equation to Adjudicate Terminal States in Tangled - Make it stand out From left to right: Graph 1, two vertices, 1 edge; graph 2, 3 vertices, 3 edges; graph 3, 4 vertices, 6 edges; graph 4, 6 vertices, 6 edges; graph 5 (Petersen graph), 10 vertices, 15 edges; and finally graph 6, the Tesseract graph, 16 vertices, 32 edges. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/5af87ef3-e31d-4026-8537-be5a899189fe/Screenshot+2024-06-15+222219.png TANGLED - The Computational Cost of Using the Schrodinger Equation to Adjudicate Terminal States in Tangled - Make it stand out Runtimes for evaluating one terminal state for the graphs above. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/0a516325-66d0-4e17-980b-2ae908bfc469/Screenshot+2024-06-15+220519.png TANGLED - The Computational Cost of Using the Schrodinger Equation to Adjudicate Terminal States in Tangled - Make it stand out Same data in graph form, with an exponential trend line overlaid. https://www.snowdropquantum.com/tangledblog/building-a-game-playing-ai-for-tangled monthly 0.5 2024-09-27 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/551fa510-86d6-41e9-beb3-3ef446de280b/Screenshot+2024-05-12+192232.png TANGLED - Building Three Tangled Agents - Make it stand out Left: a sequence of moves, shown as (top) state changes where the most recent change is shown in purple and (bottom) the game board for those states. Right: the states, ordered from beginning (top) to end (bottom), as a matrix. The actions selected here are, in order, 4, 9, 0, 2, and 6; so the red agent returned action 4 from state [0,0,0,0,0,0]; action 0 from state [0,0,0,2,0,1] and action 6 from state [1,0,2,2,0,1]. The blue agent returned action 9 from state [0,0,0,2,0,0] and action 2 from state [1,0,0,2,0,1]. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/4dc894a0-641a-4261-84b1-d58ecc918be5/MCTS_Algorithm.png TANGLED - Building Three Tangled Agents - Make it stand out The MCTS algorithm has four phases that repeat until you run out of time and/or patience. The key step to highlight here is the third one (rollout). In this phase, some policy (e.g. random choice) is applied to both players, starting from some initial state (the dark circle) and ending at that game’s terminal state. The terminal state is evaluated (win/lose/draw) and then in phase four the result is backpropagated up the chain to make it more likely that actions resulting in wins for the AI currently taking an action are more likely to be taken, and actions resulting in losses are less likely to be selected. If evaluating the terminal state is hard, MCTS gets bottlenecked by this third phase. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/5a619f8c-63e3-4d36-b0c3-8e117c17dc92/Screenshot+2024-05-12+150809.png TANGLED - Building Three Tangled Agents - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/a171e12b-70e0-490f-81a4-f1d93c6c46da/Screenshot+2024-06-16+130052.png TANGLED - Building Three Tangled Agents - Make it stand out Results of random vs MCTS agents on the three-vertex three-edge graph. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/c9eeb92f-3b89-427a-98ec-67cba1df532f/Screenshot+2024-07-14+144518.png TANGLED - Building Three Tangled Agents - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/54e9a4d9-9cc4-4d5a-9a9e-57a3e8b7eb08/Screenshot+2024-05-12+141308.png TANGLED - Building Three Tangled Agents - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/8373b07b-641f-4e62-bbb2-ede7886e3ae0/Screenshot+2024-06-16+144304.png TANGLED - Building Three Tangled Agents - Make it stand out Results of random vs MCTS agents on the four vertex six edge graph. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1afe72dd-cfc8-4a63-9d51-6ffc9f4a6f8a/Screenshot+2024-07-14+150406.png TANGLED - Building Three Tangled Agents - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/tangled-on-a-4-vertex-graph monthly 0.5 2024-09-27 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/b50e6f16-c566-43b8-bde2-b8942a18e4de/Screenshot+2024-05-12+141308.png TANGLED - Tangled on a 4-vertex graph - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d9d25350-e748-4e22-9743-d75cd929ee58/Screenshot+2024-05-12+141634.png TANGLED - Tangled on a 4-vertex graph - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/b056e54b-cc60-4b17-a148-441a45857672/Screenshot+2024-05-12+141950.png TANGLED - Tangled on a 4-vertex graph - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/tangled-on-a-three-vertex-graph monthly 0.5 2024-09-27 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d551d53f-45a2-47aa-9d1e-dea67c1751cd/Screenshot+2024-05-12+150809.png TANGLED - Tangled on a 3-vertex graph - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/c1bc5f42-8b66-4bf0-a677-5d5391f6d988/Screenshot+2024-05-12+150954.png TANGLED - Tangled on a 3-vertex graph - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/9a6c48af-412d-433d-aeeb-f3c83a1b0068/Screenshot+2024-05-12+151140.png TANGLED - Tangled on a 3-vertex graph - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/69815834-cb9b-437e-9088-37c5fd3032ff/Screenshot+2024-09-10+103455.png TANGLED - Tangled on a 3-vertex graph - Blue Wins: [[0, 1, 2, 1, 2, 1], [0, 1, 2, 1, 2, 3], [0, 1, 2, 3, 1, 1], [0, 1, 2, 3, 1, 3], [0, 1, 2, 3, 2, 1], [0, 1, 2, 3, 2, 2], [0, 1, 2, 3, 2, 3]] Red Wins: [[0, 1, 2, 1, 3, 1], [0, 1, 2, 1, 3, 3], [0, 1, 2, 2, 1, 1], [0, 1, 2, 2, 1, 3], [0, 1, 2, 2, 3, 1], [0, 1, 2, 2, 3, 2], [0, 1, 2, 2, 3, 3]] https://www.snowdropquantum.com/tangledblog/play-the-game-8e7fb monthly 0.5 2025-01-07 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - An Introduction to Tangled - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - An Introduction to Tangled - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/6808f528-6411-4177-a349-9be97b675cf9/Screenshot+2024-05-11+072018.jpg TANGLED - An Introduction to Tangled - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/91fa590a-358b-42f2-9cac-99570bce66b3/Screenshot+2024-05-11+074241.jpg TANGLED - An Introduction to Tangled - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/727ee286-e2ea-4b8e-9d94-d41bcd963b09/Screenshot+2024-05-11+074404.jpg TANGLED - An Introduction to Tangled - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/65281c24-6856-45c9-a7cb-39c35f8b750c/Screenshot+2024-05-11+074704.jpg TANGLED - An Introduction to Tangled - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/4af2cc47-7d39-4ae3-943c-539fa0e652a4/Screenshot+2024-06-12+154418.jpg TANGLED - An Introduction to Tangled - A finished Tangled game played on the Petersen graph. The Petersen graph is a type of graph called a snark, which I spent wayyyy too long reading about. Also it contains what Bill Macready once called a ‘witch thing’ while making air quotes, which still makes me laugh. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/8c0765c2-3ce1-456f-9a24-f628b84e113c/Screenshot+2024-06-12+153748.jpg TANGLED - An Introduction to Tangled - A finished Tangled game played on a 4D tesseract graph. Tesseracts are cool! https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/a6611992-476c-4189-8080-90b48d19b8b9/equation+%2827%29.png TANGLED - An Introduction to Tangled - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/aa8227c6-bd87-40ca-a7f7-8e09a3391855/equation+%289%29.png TANGLED - An Introduction to Tangled - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/quantum-mechanics-and-quantum-computers monthly 0.5 2024-09-23 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/3aeac875-7157-469f-b2f4-24aeaac560b6/equation+%2832%29.png TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/9d8ec416-779e-472d-9da0-aa1fd339fdc5/equation+%2833%29.png TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/5716dc42-2622-47cb-a701-86ff47615423/equation+%2834%29.png TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/6e4f5170-9b10-40d3-9d6b-480072d69656/equation.png TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1138bc3f-5528-480d-9c13-3b339dab9b56/Screenshot+2024-05-13+182605.png TANGLED - Quantum Mechanics and Quantum Computers - The Advantage2_prototype2.3 fast annealing schedule for the D-Wave Advantage2.3 system. You can find the data files for these terms here. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/524285fd-05c4-40dd-9977-d8eac5112b44/equation+%2835%29.png TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/579b73c2-1556-498a-9c9f-ac44828767d7/equation+%2837%29.png TANGLED - Quantum Mechanics and Quantum Computers - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/influence monthly 0.5 2024-09-05 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/53558fef-35cd-4a13-a852-389ea604be01/Designer+%288%29.jpeg TANGLED - Influence - In models of magnetism, what I’m calling influence is called a spin-spin correlation function. You can call it that if you want. Yeah I know I shouldn’t be renaming things that already have perfectly good names. Sue me. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - Influence - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/17807dc8-5085-4a29-9e31-7e13521ed282/equation+%287%29.png TANGLED - Influence - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - Influence - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/3a8b4e5c-8491-48d5-8e91-fc858e8a72ff/Designer+%289%29.jpeg TANGLED - Influence - A remarkably non-accurate image of Gary, Oprah, Coco, and Margaret. I’ll leave it up to you to decide if that’s stud Gary or stupid Gary. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - Influence - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - Influence - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/the-ising-model-and-an-introduction-to-sampling-from-a-probability-distribution monthly 0.5 2024-09-04 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/8096d3d0-f207-4a20-80c0-de3e6a302474/Designer+%287%29.jpeg TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - The Universe appears to be made out of a small number of patterns, where all the apparent complexity of the world comes from them combining in different ways. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/edda570e-9f60-4c30-bc35-2c0e8daba2ac/eric-weinstein-3375805944.png TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - See it’s not so hard being a theoretical physicist!!! This sort of thing is basically all we do. Except if you’re Eric Weinstein, in which case you also do a lot of podcasts. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/a6611992-476c-4189-8080-90b48d19b8b9/equation+%2827%29.png TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/920f2b46-d686-4115-a658-4f851d692325/Screenshot+2024-09-04+104903.png TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - Left: a visualization of the N=3 Ising model instance as a 3-vertex complete graph, with h and J terms as labels on vertices and edges respectively. Right: an enumeration of all 2³=8 possible states and their energies, calculated by using the Ising model equation. For the graph state, I colored the vertices red for s=+1 and green for s=-1, because I thought it looked cheery! You can see that for this instance, the lowest energy state is s₀=-1, s₁=+1, and s₂=-1 with an energy of -2.7. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/6347191a-89cd-4edc-b250-a48f9523a935/equation+%2828%29.png TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/200a7d4e-69fa-40a1-b6ad-c4505d633bef/equation+%2829%29.png TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e91eac7e-9b96-4118-866b-6ad41eb3555d/Screenshot+2024-09-04+113038.png TANGLED - The Ising Model, the Boltzmann Distribution, and What Sampling From a Probability Distribution Means - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/correlations monthly 0.5 2024-09-01 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e6135a02-4768-4b74-a955-b330819fa31e/Designer+%286%29.jpg TANGLED - Correlation!!! - Give it a try! Get a friend, split your poker chips with them, break out your Cannda and Untd States coins, and start tossing them! See how long it takes before one of you wins all the money! https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/7e3135f6-5bde-4fc4-96b4-189ef961974d/Designer+%283%29.jpg TANGLED - Correlation!!! - Throw this AI generated coiny-stick business and you either get both heads or both tails on every throw. The result of our coin toss is now perfectly correlated! This ridiculous thing produces a correlation between coin 0 and coin 1 of C₀₁ = +1. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/94a91fc7-f758-4de0-87a5-c7a4fa1ee523/Designer+%284%29.jpg TANGLED - Correlation!!! - Throw this coiny-stick business and you always get one heads and one tails. The result of our coin toss is now perfectly anti-correlated. This thing produces a correlation between coin 0 and coin 1 of C₀₁ = -1 (perfect anti-correlation). https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/9e04de30-3907-49b2-9915-9bd12250a635/PXL_20240901_195730591.jpg TANGLED - Correlation!!! - Yeah, I actually flipped both of these coins 10 times and got the results in the above list. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d7058897-3dcb-4db2-ab24-8a691f1d6d05/equation+%2821%29.png TANGLED - Correlation!!! - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/the-thread-im-about-to-pull monthly 0.5 2024-09-01 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/b6be2250-6132-47c0-8f05-32c7ed93fabb/800px-Alfonso_LJ_97V.jpg TANGLED - The Thread I’m About to Pull - The game of astronomical tables, from the illuminated manuscript Libro de los juegos (1334). “Indeed, a person who has become habituated to gaming with pigeons, playing chess, or gambling, so that it becomes second-nature to him, will give all the comforts of the world and all that he has for those (pursuits) and cannot keep away from them.” — The Alchemy of Happiness (c. 1100). https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - The Thread I’m About to Pull - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/00321d2b-b7da-4d95-948c-6aaa3b10eafb/dT9rEGo6c.jpg TANGLED - The Thread I’m About to Pull - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://www.snowdropquantum.com/tangledblog/quantum-computers-should-also-be-useful monthly 0.5 2024-08-30 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/b26c7551-bf83-4fb9-adab-de1645ca8b47/Designer+%281%29.jpeg TANGLED - Quantum Computers Should (Also) Be Useful - Make it stand out The quantum computing field has been around for 25+ years and we still don’t have a single use case where it makes sense to use a QC vs the alternatives. Yes, QCs are amazing testbeds for fundamental science. But they should also be useful! And right now, they … aren’t. At least not in the clearcut way we all want them to be. https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/84c69984-96b9-46ee-9baa-c1413691dea4/Screenshot+2024-08-30+111356.png TANGLED - Quantum Computers Should (Also) Be Useful - Make it stand out The world is constantly computing solutions to the laws of physics, and it’s really good at it! https://www.snowdropquantum.com/tangledblog/hello-world monthly 0.5 2024-11-19 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/e76f85ee-707a-460b-b8f3-bc2ed0192552/snowdrops.jpg TANGLED - Hello World! - Make it stand out These are snowdrops. They symbolize new beginnings, hope, rebirth, and the ability to overcome challenges. https://www.snowdropquantum.com/hex daily 0.75 2026-01-25 https://www.snowdropquantum.com/foundations-1 daily 0.75 2026-03-31 https://www.snowdropquantum.com/sereno-sofia-parazi-photography daily 1.0 2026-01-17 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/1712516691047-K55ZP5QV3B9VKQI29EDN/LOGO_ONLY.png https://www.snowdropquantum.com/about daily 0.75 2026-02-03 https://www.snowdropquantum.com/mission daily 0.75 2024-08-29 https://www.snowdropquantum.com/contact daily 0.75 2024-08-29 https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/d66a96a0-fffc-4ba1-8086-8c2d64bfaea0/Designer+%285%29.jpeg CONTACT https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/c9c3a090-a149-4393-a745-222fa9dce210/Designer+%284%29.jpeg CONTACT https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/6e0e9cd7-e7fe-49f4-96e5-4b301c61c7fb/Designer+%282%29.jpeg CONTACT https://images.squarespace-cdn.com/content/v1/660c6f49bb6d3201aae1d771/255f46aa-392c-4a66-956c-872184243403/Designer+%283%29.jpeg CONTACT