Meeting notes 11 07 25

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David: Thinking about modules, and their possible use in quantum computing.

Aram: Just worked on writing up old results.

Melanie: Reading about welded trees.

Tom: Wrote code to, given a psd matrix M, calculate <math>\max\{\text{tr} M\rho: \rho \in k\text{-extendable}\}</math>. This reduces to computing <math>\|P(M^{AB_1}\otimes I^{B_2\ldots B_k})P\|</math>, where P projects onto the symmetric subspace of <math>B_1,\ldots,B_k</math>.

One question is how to construct P? It projects onto a space of dimension <math>d\binom{k+d-1}{d-1}</math>.

Showed some numerical results. Discussed how to construct the basis. This part takes most of the time. Improvements (sparsity, implicit representation of matrices) were considered.

Rowan: Looking at quick matrix-vector multiplies. Also looking at generating Hamiltonians for random 2-SAT. A somewhat easier alternative is to look at planted solutions.

Greg: Given a collection of measurements, create a code which has stabilizers, gauge qubits and logical qubits. Of these, the stabilizers and gauge operators generate the set of measurements (with stabilizers defined to be the subset that commutes with everything else).

Why is this better? Consider the 5-qubit code, with stabilizer generators XZZXI, IXZZX, XIXZZ, ZXIXZ. These have weight 4. By contrast, Bacon-Shor requires only 2-qubit measurements.

Does there exist an 8-qubit code with 2-qubit measurements? Naive exhaustive search involves a space of size 2256, but symmetry can significantly reduce this.

Johnny: Talking about reducing 3-local Hamiltonians to 2-local Hamiltonians.

Magic theorem: Assume that H has gap <math>\Delta</math> around <math>\lambda_*</math>. Thus if <math>\lambda_\pm = \lambda_* \pm \Delta/2</math>, then we assume the spectrum of H is contained in <math>[-\infty,\lambda_-]\cup [\lambda_+,\infty]</math>. Assume <math>\|V\|\leq \Delta/2</math>. Then for any <math>\epsilon>0</math> there exists Heff with spectrum contained in [c,d] with <math>c<d<\lambda_*-\epsilon</math>. If <math>\|\Sigma_-(z) - H_{\rm eff}\|\leq \epsilon</math> for all <math>z\in[c-\epsilon,d+\epsilon]</math> then every eigenvalue <math>\tilde\lambda_j</math> of <math>\tilde H_{\tilde{\cal L}__-}</math> is epsilon-close to <math>\lambda_j</math> of Heff.

The gadget: Let B1, B2, B3 be single-qubit operators. Let <math>H = -\frac{\delta^{-3}}{4} I \otimes (\sigma_1^z\sigma_2^z + \sigma_2^z\sigma_3^z + \sigma_1^z\sigma_3^z)</math> <math>V = Y\otimes I +\delta^{-1}(B_1^2 + B_2^2 + B_3^2)\otimes I -\delta^{-2}(B_1\otimes \sigma_1^x + B_2 \otimes \sigma_2^x + B_3 \otimes \sigma_3^x)</math> where Y is some 2-local interaction to be determined later.

To analyze this, observe that <math>{\cal L}_-</math> is spanned by <math>\{|000\rangle, |111\rangle\}</math> and we see that <math>\Sigma_-(z)</math> is dominated by a term of the form <math>-6 B_1B_2B_3 \otimes \sigma_{\rm eff}^x</math>.