Overlapping qubits THOMAS VIDICK CALIFORNIA INSTITUTE OF TECHNOLOGY J O I N T WO R K W I T H BEN REICHARDT, UNIVERSITY OF SOUTHERN CALIFORNIA

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1 Overlapping qubits THOMAS VIDICK CALIFORNIA INSTITUTE OF TECHNOLOGY JOINT WORK WITH BEN REICHARDT, UNIVERSITY OF SOUTHERN CALIFORNIA

2 What is a qubit? Ion trap, Georgia Tech Superconducting qubit, Martinis group Photonic qubit, TU Delft Quantum dot, JQI

3 What is a qubit? 2

4 What is a qubit? a qubit

5 What is a qubit? Ion trap, Georgia Tech Superconducting qubit, Martinis group Photonic qubit, TU Delft Quantum dot, JQI

6 What is a qubit? Definition: a qubit is a pair of anti-commuting reflections. X = X, Z = Z X 2 = Z 2 = I X, Z = 0 Theorem 1: (X, Z) anti-commuting reflections on H H C 2 H? H C 2 H, X σ X I, Z σ Z I Proof. Easy argument using Jordan s lemma: (X, Z) induce decomposition of H in 2-dimensional planes

7 What are n qubits? 5 superconducting qubits, IBM 5 Xmon qubits, Martinis group 128 qubit chip, D-wave 16 atomic ions, Monroe group

8 What are n qubits? C 2 C 2 C 2 Def: n independent qubits X 1, Z 1, X 2, Z 2,, (X n, Z n ): X i, X j = Z i, Z j = X i, Z j = Z i, X j = 0 for all i j Theorem 2: (X i, Z i ) independent qubits H C 2 C 2 H, X i σ i X I, Z i σ i Z I Proof. Easy inductive argument using commutation to separate the qubits.

9 Noisy qubits? Real systems are never perfect and even if they were, we couldn t tell! What is a noisy qubit? X, Z ε? n noisy qubits? X i, Z j ε? How close are noisy qubits to real qubits? Do n noisy qubits require 2 n dimensions? Are there good experimental tests for noisy qubits?

10 Outline 1. Defining qubits 2. Noisy qubits: operator norm a. Lower bound: packing noisy qubits b. Upper bound: separating noisy qubits 3. Noisy qubits: state-dependent norm a. Testing noisy qubits (i) b. Faking perfect qubits c. Testing noisy qubits (ii)

11 Outline 1. Defining qubits 2. Noisy qubits: operator norm a. Lower bound: packing noisy qubits b. Upper bound: separating noisy qubits 3. Noisy qubits: state-dependent norm a. Testing noisy qubits (i) b. Faking perfect qubits c. Testing noisy qubits (ii)

12 Noisy qubits: operator norm Def.: noisy qubit = reflections (X, Z) s.t. {X, Z} ε Geometrically: Lemma: if X, Z satisfy {X, Z} ε there exists X, Z dihedral angles s.t. X, Z ε = 0 and X X, Z Z O(ε) or π A noisy qubit ε 2 is never far from a perfect qubit Def.: qubits X 1, Z 1 and X 2, Z 2 ε-overlap if P 1, Q 2 ε for all P, Q {X, Z}

13 Noisy qubits: operator norm Q: How many ε-overlapping qubits fit in 2 n dimensions? Maximum m such that X 1, Z 1,, X m, Z m reflections in C 2n, {X i, Z j } ε and [X i, Z j ] ε for i j 1. n ε n 3. n 1+ε ε n any number, as long as ε 2 n 6. any number, as long as ε 1/π A: 1 2 eε2 n/8

14 Lower bound: packing noisy qubits 1. Johnson-Lindenstrauss lemma: 2 n orthogonal unit vectors in 2 n dimensions 2 n ε-orthogonal unit vectors in n/ε 2 dimensions 2. Tsirelson s Clifford embedding: real unit vector u R d observable A u C 2d 2 d s.t. {A u, A v } = u v Id C 2 d The construction: 1. (J-L) 2 n vectors u i s.t. u i u j ε d = n/ε 2 2. (Tsirelson) 2 n obs. A i s.t. A i, A j ε d = 2 n/2ε2 3. Noisy qubits: X j = ia 2j A 2j+1 Z j = ia 2j+1 A 2j+2

15 Upper bound: separating qubits Theorem 2: X 1, Z 1,, X n, Z n ε-overlapping qubits independent qubits (X i, Z i ) s.t. X i X i 8n ε Corollary: ε 1 n d 2n Theorem 1: ε C logn n d 2 n1/c Open: 1 n ε C log n n?

16 Upper bound: separating qubits Theorem 2: X 1, Z 1,, X n, Z n ε-overlapping qubits independent qubits (X i, Z i ) s.t. X i X i 8n ε Proof: sequential block-diagonalization P 1 P 1 = P 1 P 2 P 2 = P 1 P 2 P P 1 P 2 1 P 1 P 3 P 3 = P 1 P 3 P P 1 P 3 (1 P 1 ) Need careful control of error blow-up! P i, P j ε P i, P j 2ε P i, P j 4ε

17 Upper bound: separating qubits Theorem 2: X 1, Z 1,, X n, Z n ε-overlapping qubits independent qubits (X i, Z i ) s.t. X i X i 8n ε Example: Bound is tight up to the constant Separating n noisy qubits can take linear effort 1. X 1, Z 1,, X n, Z n perfect qubits. 2. Set H = (Z Z n/2 )(Z n/2+1 + Z n ) 3. Evolve X n/2+1, Z n/2+1,, X n, Z n (X i, Z i ) = e iεh X i, Z i e iεh

18 Outline 1. Defining qubits 2. Noisy qubits: operator norm a. Lower bound: packing noisy qubits b. Upper bound: separating noisy qubits 3. Noisy qubits: state-dependent norm a. Testing noisy qubits (i) b. Faking perfect qubits c. Testing noisy qubits (ii)

19 Testing (noisy) qubits Quantities such as P, Q are not experimentally accessible! State ψ, projections P, Q, observable quantities ψ P ψ, ψ PQP ψ, estimate state-dependent norm P, Q ψ 1) How do the packing/separating qubit theorems adapt? Are state-dependent bounds sufficient? 2) Are there effective tests for a noisy n-qubit system?

20 1) Testing (noisy) qubits assuming two systems X 1, Z 1 X 1, Z 1 (i) Testing anti-commutation CHSH inequality: ψ A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 ψ {A 0, A 1 } ψ ε H = H A H B 2 2 ε (ii) Testing commutation Dummy input test: A gets (i, 0/1); B gets (i, 0/1) and (j, 0/1), unordered [A ia, A ja ] ψ ε for i j, a, a {0,1}

21 1) Testing (noisy) qubits with only one system? X 1, Z 1,, X n, Z n (i) Testing anti-commutation Measure X Evolve using U = e iπ 2 X e iπ 2 Z e iπ 2 X e iπ 2 Z = XZXZ Measure X, check consistency {X, Z} ψ ε H (ii) Testing commutation Measure X i Measure X j or Z j Measure X i, check consistency [X i, Z j ] ψ ε for i j

22 1) Testing (noisy) qubits with only one system? X 1, Z 1,, X n, Z n (i) Measure X, evolve using U = XZXZ, measure X (ii) Measure X i, measure X j or Z j, measure X i Pass both tests perfectly in ~n 3 dimensions! X 1, Z 1 X 2, Z 2 X n, Z n X X 1, Z 1 X 3, Z 3 X n, Z 2, Z 2 n

23 1) Testing (noisy) qubits with only one system? X 1, Z 1,, X n, Z n (i) Measure X, evolve using U = XZXZ, measure X (ii) Measure X i, measure X j or Z j, measure X i Pass both tests perfectly in ~n 3 dimensions! Solution: consider longer tests (ii) Measure X 1, Z 1,, X i, Z i,, X n, Z n, Z i or Z 1, X 1,, Z i, X i,, Z n, X n, X i Theorem: Success 1 ε kn ε-indist. from n-independent qubit-system for all single-qubit unitary circuits of depth k

24 2) Testing (noisy) qubits with only one system? X 1, Z 1,, X n, Z n (i) Measure X, evolve using U = XZXZ, measure X (ii) Measure X 1, Z 1,, X i, Z i,, X n, Z n, Z i or Z 1, X 1,, Z i, X i,, Z n, X n, X i Theorem: Success 1 ε kn ε-indist. from n-independent qubit-system for all single-qubit unitary circuits of depth k Proof. Introduce ancilla EPR pairs to simulate fresh qubits Use noisy qubit operators to build SWAP gate between noisy and fresh qubits Corollary: Success 1 ε dim H 1 n 4 ε 2 n

25 Summary Qubit: pair of anti-commuting reflections X, Z Independent qubits: (X 1, Z 1 ) and (X 2, Z 2 ) s.t. X 1, Z 2 = = 0 Can independent qubits be tested? Can tests be ε-fooled in 2 n dimensions? poly(n) dimensions for ε 1/ n, operator norm poly(n) dimensions for ε = 0, state-dependent norm Bounds are not all tight More efficient state-dependent test? Testing circuits with two-qubit unitaries Testing for (single-prover) delegated computation?