Stage 5: (Advanced) YBE-powered compression#

Finally, we implement the YBE-powered quantum circuit compression scheme for the linear-depth Trotterization circuits constructed in Stage 3. Under certain conditions on the coupling constants, we will obtain circuits whose depth is totally independent of the Trotter number!

This means we can approximate the time propagator $U(t)$ to arbitrary accuracy without paying any cost in terms of added circuit depth. The catch is that we must be willing to expend some (classical) computation up front in order to find appropriate parameters for the compressed circuit.

The approximation scheme relies on two circuit identities, known as reflection symmetry and merge in Peng et al. [PGAG22]. The first is a YBE; when at least one of the coupling constants is zero, there exist parameters $\theta', \theta''$, and $\theta'''$ such that, for instance,

\[B_j(\theta) B_{j+1}(\theta) B_j(\theta) = B_{j+1}(\theta') B_j(\theta'') B_{j+1}(\theta'''),\]

for any given initial parameter choice. The figure below illustrates the so-called reflection symmetry.

Reflection symmetry identity satisfied by the block operators when $J_x \cdot J_y \cdot J_z = 0$. — Reflection symmetry identity satisfied by the block operators when $J_x \cdot J_y \cdot J_z = 0$#

The so-called merge identity allows for merging two blocks in acting on the pair of sites into one; in particular,

\[B_j(\theta) B_j(\theta') = B_j(\theta + \theta')\]

for any $\theta, \theta' \in \mathbb{R}^3$.

When combined, these identities explain the circuit compression scheme. For instance, if we consdier $N = 3$ sites and use $n = 2$ Trotter steps, we can compress the Trotterized circuit as follows:

\[\begin{split}\begin{align*} U(t) &\approx \big(B_1(\theta) B_2(\theta) \big)^2 \\ &= \big(B_1(\theta) B_2(\theta)\big) \big(B_1(\theta) B_2(\theta)\big) \\ &= \big(B_1(\theta) B_2(\theta) B_1(\theta)\big) B_2(\theta) \\ &= \big(B_2(\theta') B_1(\theta'') B_2(\theta''')\big) B_2(\theta) \\ &= B_2(\theta') B_1(\theta'') B_2(\theta''' + \theta), \end{align*}\end{split}\]

The problem is finding the new parameters $\theta'$, $\theta''$, and $\theta'''$. For simplicity, we will assume a black-box algorithm for to complete this task: simply use the provided src.circuit_compressor.get_optimized_params() method.

The compression scheme can be extended to longer spin chains. For instance, the figure below, taken from Peng et al. [PGAG22], illustrates how to compose a sequence of symmetry reflections to produce the mirror identity that can be used to compress circuits with $N = 4$ qubits.

A sequence of reflection symmetries on 3 sites used to construct the so-called identity on $N = 4$ qubits.#

The following figure, also taken from Peng et al. [PGAG22], illustrates how to combine the mirror and merge identities to produce a compression scheme for $N = 4$ lattice sites.

Combining the mirror (red braces) and merge identity (black dotted box) to compress a pair of columns in the Trotterized evolution circuit on $N = 4$ sites.#

It turns out that for any number of Trotter steps $n$, you can always compress the Trotterized evolution circuit on $N$ sites down to a quantum circuit with just $N (N-1) / 2$ block operators. You can prove this below in the algebraic formulation section.

Constant-depth(!) time evolution under certain Heisenberg Hamiltonians#

For the rest of this section, assume $J_y = h = 0$, so we will focus on the $H_x + H_z$ Hamiltonian.

Now implement the YBE-powered compression scheme for $N = 3, 4$ sites using the “moves” illustrated in the figures above. Do this in three steps: first implement the CircuitCompressor.get_ybe_update() method, then the CircuitCompressor.get_mirror_update() method, and finally proceed to the CircuitCompressor.compress_circuit() method.

Hint: Label the blocks in the circuit and identify which sets of parameters are involved in each symmetry application.

class src.circuit_compressor.CircuitCompressor(xyz_evolution_qc)[source]#

A class that implements the YBE-powered QTD circuit compression scheme.

compress_circuit()[source]#

Compress this time evolution circuit using the YBE.

OUTPUT:

Returns a compressed XYZEvolutionCircuit object with $N (N - 1)/2$ blocks that is equivalent to the uncompressed Trotterization circuit self.deep_qc.

get_mirror_update(params, l2r=True)[source]#

Update the $4$-qubit time propagator circuit parameters following the so-called mirror step.

INPUT:

params – a (6, 2) NumPy array describing the parameters for each of the six blocks

l2r – a boolean indicating whether we are applying the symmetry left-to-right or vice versa

OUTPUT:

A (6, 2) NumPy array describing the parameters of the mirrored circuit.

get_ybe_update(params, l2r=True)[source]#

Update $3$-qubit time propagator circuit parameters according to the Yang-Baxter Equation (YBE).

INPUT:

params – a (3, 2) array describing the parameters for each of the three blocks

l2r – a boolean indicating whether we are applying the symmetry left-to-right or vice versa

OUTPUT:

A (3, 2) NumPy array describing the parameters of an equivalent circuit.

Exercise 5.1

Now modify the xyz_evolution_challenge.ipynb notebook so it also computes the staggered magnetization $M_{\mathrm{stag}}(t)$ using the compressed circuits for $N = 3$ and $N = 4$ sites. Comment on your results and include your plots showing four magnetization curves on the same set of axes for each $N$.

Challenge!

Use the reflection symmetry and the merge identity to design and implement a compression scheme for circuits on $N > 4$ lattice sites. Report the complexity of your algorithm as the number of reflection symmetries required.

(Expert) Algebraic formulation#

To conclude, we prove the following theorem:

Suppose the block operators $B_j$ satisfy the YBE. Then any Trotterized evolution circuit constructed considered here, using any number of block operators, may be expressed as an equivalent circuit with depth at most $N$ and using at most $N(N-1)/2$ blocks.

We consider a parameter-independent version of the block operators by simply forgetting they are parametrized. However, we do not lose generality; we may easily extend the results to the parameter-dependent case by appropriately updating the parameters in each block after each symmetry application.

Under the YBE hypothesis, the block operators generate a representation of the Artin braid group $\mathcal{B}_n$, which is generated by $b_1, \ldots, b_n$ subject to the braid relations:

\[b_j b_{j+1} b_j = b_{j+1} b_j b_{j+1}, \quad\text{and}\quad b_j b_k = b_k b_j \text{ whenever } |k - j| > 1.\]

Incidentally, the topological quantum computation model is also based on unitary representations of the braid group Aboumrad [Abo22].

In addition to the braid relations, the block operators satisfy $B_j(\theta)B_j(\theta') = B_j(\theta + \theta')$, which becomes $B_j^2 = B_j$ in the parameter-independent version.

Therefore, we can state more precisely that the block operators generate a representation of the $0$-Hecke algebra. The Iwahori-Hecke algebra $H_n(q)$ is generated by $T_1, \ldots, T_{n-1}$ subject to the braid relations together with the quadratic relation

\[T_j^2 = (q - 1)T_j + q.\]

Thus if we set $q = 0$ every generator becomes idempotent: $T_j^2 = T_j$.

The Hecke algebra is well-known and well-studied; it is closely related to the symmetric group and it appears in a variety of places.

Exercise 5.2

Use Matsumoto’s Monoid Lemma or an explicit basis of the (finite-dimensional) $0$-Hecke algebra to prove our compression scheme existence theorem.