Overlapping qubits. Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH. The parallel-repeated magic square game is rigid

Transcription

1 Rui Chao USC Ben W. Reichardt USC Overlapping qubits Chris Sutherland USC Thomas Vidick Caltech arxiv Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH Andrea W. Coladangelo Caltech arxiv The parallel-repeated magic square game is rigid Matthew Coudron MIT Anand Natarajan MIT arxiv

2 Rui Chao USC Ben W. Reichardt USC Overlapping qubits Chris Sutherland USC Thomas Vidick Caltech arxiv Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH Andrea W. Coladangelo Caltech arxiv The parallel-repeated magic square game is rigid Matthew Coudron MIT Anand Natarajan MIT arxiv

3 5 superconducting qubits, IBM 16 trapped ion qubits, UMD/NIST 1152 superconducting qubits, D-Wave quantum computers are scaling up n qubits 2 n dimensions exponentially hard to analyze

4 5 superconducting qubits, IBM 16 trapped ion qubits, UMD/NIST 1152 superconducting qubits, D-Wave quantum computers are scaling up n qubits 2 n dimensions exponentially hard to analyze small state/process tomography How to test quantum computers? medium error correction? small simulation? our tests! large factorization

5 box box Testing quantum systems -Is it quantum? -How many qubits? -How much entanglement? -How does it work? box classical Accept or Reject?

6 box box Testing quantum systems -Is it quantum? -How many qubits? -How much entanglement? -How does it work? box classical Accept or Reject? Goal: tests that and(or) and(or) for large quantum systems take polynomial time scalability efficiency with high probability completeness & soundness tolerate constant noise robustness & rigidity

7 my part: next: Test the dimensionality of a single quantum system How many qubits ^ overlapping Test the number of (tilted) EPR pairs between two systems How much entanglement Andrea: using tilted CHSH games Matthew: using Magic Square games

8 Quantum systems are made of qubits in tensor product n qubits 2 n dim

9 Quantum systems are made of qubits in tensor product n qubits 2 n dim In general qubits can overlap operations on one qubit can slightly affect the others

10 Quantum systems are made of qubits in tensor product n qubits 2 n dim In general qubits can overlap operations on one qubit can slightly affect the others ε U 1 U 2 k[u 1,U 2 ]kapple

11 Quantum systems are made of qubits in tensor product n qubits 2 n dim In general qubits can overlap operations on one qubit can slightly affect the others ε n ε-overlapping qubits U 1 U 2 k[u 1,U 2 ]kapple n 1/ ε 2 dim

12 Theorem 1: n overlapping qubits can fit in poly(n) dimensions ε-overlap (operations on one qubit can affect any other qubit by at most ε) n 1/ ε 2 dimensions

13 Theorem 1: n overlapping qubits can fit in poly(n) dimensions ε-overlap (operations on one qubit can affect any other qubit by at most ε) n 1/ ε 2 dimensions Theorem 2: Given access to n (overlapping) qubits, a test s.t. Pr[pass test] 1-ε dimension (1-O(n 2 ε)) 2 n

14 Definitions: A qubit in Indeed: H is a pair of anti-commuting reflections on it H ' C 2 H 0 {X, Z} =0) X ' x 1 Z ' z 1

15 Definitions: A qubit in Indeed: H is a pair of anti-commuting reflections on it H ' C 2 H 0 {X, Z} =0) X ' x 1 Z ' z 1 The overlap ε of 2 qubits (X1,Z1), (X2,Z2) in max k[p 1,Q 2 ]k P,Q2{X,Z} H is given by

16 Definitions: A qubit in Indeed: H is a pair of anti-commuting reflections on it H ' C 2 H 0 {X, Z} =0) X ' x 1 Z ' z 1 The overlap ε of 2 qubits (X1,Z1), (X2,Z2) in max k[p 1,Q 2 ]k P,Q2{X,Z} ε=0 qubits in tensor product: H is given by X 1 ' x I 1 Z 1 ' z I 1 X 2 ' I x 1 Z 2 ' I z 1

17 Theorem 1: n ε-overlapping qubits can fit in n Ω (1/ε 2 ) -dimensional Hilbert space. Proof idea: nearly orthogonal vectors 3n points in R O log n 2

18 Theorem 1: n ε-overlapping qubits can fit in n Ω (1/ε 2 ) -dimensional Hilbert space. Proof idea: nearly orthogonal vectors group in threes nearly orthogonal subspaces e1 g1 f1 en gn fn

19 Theorem 1: n ε-overlapping qubits can fit in n Ω (1/ε 2 ) -dimensional Hilbert space. Proof idea: nearly orthogonal vectors group in threes nearly orthogonal subspaces Clifford algebra rep. nearly commuting qubits e1 g1 f1 X = i E F en gn (n Ω(1/ ε 2 ) -dim ref.) fn Z = i E G

20 Theorem 1: n ε-overlapping qubits can fit in n Ω (1/ε 2 ) -dimensional Hilbert space. Proof idea: nearly orthogonal vectors group in threes nearly orthogonal subspaces Clifford algebra rep. nearly commuting qubits e1 g1 f1 X = i E F en gn (n Ω(1/ ε 2 ) -dim ref.) fn Z = i E G Note: meaningful only if ε=ω( (log n/n) )

21 Dimension test: Given access to n qubits 1. Sequentially store n random qubits ( 0, 1, +, or - ) 2. Retrieve a random index & check it s correct 1 j n

22 Dimension test: Given access to n qubits 1. Sequentially store n random qubits ( 0, 1, +, or - ) 2. Retrieve a random index & check it s correct 1 j n Theorem 2: Pr[pass test] 1-ε dimension (1-O(n 2 ε)) 2 n Note: meaningful only if ε=o( 1/n 2 )

23 Summary Qubit: anti-commuting reflection pair Overlapping qubits: nearly commuting reflections Qubit packing: n overlapping qubits can fit in poly(n) dimensions Qubit separation: Pr[pass test] 1-ε dimension (1-O(n 2 ε)) 2 n

24 Summary Qubit: anti-commuting reflection pair Overlapping qubits: nearly commuting reflections Qubit packing: n overlapping qubits can fit in poly(n) dimensions Qubit separation: Pr[pass test] 1-ε dimension (1-O(n 2 ε)) 2 n Applications and open questions: Test functionality Loosen assumptions & run experiments Self-testing of EPR states

25 Summary Qubit: anti-commuting reflection pair Overlapping qubits: nearly commuting reflections Qubit packing: n overlapping qubits can fit in poly(n) dimensions Qubit separation: Pr[pass test] 1-ε dimension (1-O(n 2 ε)) 2 n Applications and open questions: Thank you! Test functionality Loosen assumptions & run experiments Self-testing of EPR states