Hamiltonian Encoding¶
A physics-motivated quantum data encoding that maps classical features into quantum states through time evolution under a data-dependent Hamiltonian. Creates rich entanglement structures via configurable two-qubit interactions.
+=====================================================================+
|| ||
|| |psi(x)> = exp(-i H(x) t) |+>^{(x)n} ||
|| ||
|| "Data shapes the Hamiltonian --> physics does the encoding" ||
|| ||
+=====================================================================+
1. The Core Idea¶
Instead of mapping features directly to rotation angles, Hamiltonian encoding treats the input data as couplings in a physical Hamiltonian and then simulates the time evolution of that system. The quantum state that emerges after evolution is the encoded representation.
Classical Data Hamiltonian Quantum State
+------------+ +---------------------+ +-------------------+
| x_0 = 0.3 |--+ | | | |
| x_1 = 0.7 |--+--->-| H(x) = SUM c(x) P |--exp-->| |psi> = e^{-iHt} |
| x_2 = 0.1 |--+--->-| data-dependent | | |+++> |
| x_3 = 0.9 |--+ | Pauli strings | | (entangled!) |
+------------+ +---------------------+ +-------------------+
| |
Features become Entanglement
coupling strengths encodes correlations
The initial state is |+>^n = H^n |0...0>, an equal superposition over
all computational basis states. This starting point gives the Hamiltonian
evolution maximal "room" to create complex quantum states.
Step 1: |0...0> --[H (x) H (x) ... (x) H]--> |+...+> (uniform superposition)
Step 2: |+...+> --[exp(-i H(x) t)]--------> |psi(x)> (entangled data state)
2. The Hamiltonian Framework¶
Mathematical Definition¶
The data-dependent Hamiltonian H(x) is a sum of Pauli operators weighted
by functions of the input features:
H(x) = SUM_j c_j(x) P_j
where:
c_j(x) = coefficient derived from input features x
P_j = Pauli string (tensor product of I, X, Y, Z)
The encoding unitary is the matrix exponential:
Trotterization¶
Since terms in H(x) generally don't commute, the exponential is approximated via first-order Trotter decomposition:
exp(-i H(x) t) ~= [ PROD_j exp(-i c_j(x) (t/r) P_j) ]^r
where r = reps (number of Trotter steps)
Accuracy: error ~ O(t^2 / r) --> more reps = better approximation
Exact Evolution First-Order Trotter (r=1) Higher-Order (r=3)
================= ========================= ===================
e^{-i(A+B+C)t} ~= e^{-iAt} e^{-iBt} e^{-iCt} [e^{-iAt/3} e^{-iBt/3} e^{-iCt/3}]^3
(intractable for (product of simple gates, (more accurate,
large systems) easy to implement) deeper circuit)
3. Four Hamiltonian Types¶
Each type creates a different interaction structure. The choice of Hamiltonian determines what correlations are captured and the resulting entanglement.
3.1 IQP (Instantaneous Quantum Polynomial)¶
Diagonal Hamiltonian with Z and ZZ terms. Creates correlations in the computational basis without basis changes.
H_iqp(x) = SUM_i x_i Z_i + SUM_{i<j} f(x_i, x_j) Z_i Z_j
Single-qubit: x_i * Z_i (local field)
Two-qubit: (pi - x_i)(pi - x_j) * Z_i Z_j (pairwise coupling)
Circuit for one Trotter step (4 qubits, full entanglement):
+---+ +----------+
|+> ---| H |--| RZ(2a_0) |--*-----------*-----------*-----------
+---+ +----------+ | | |
+---+ +----------+ | +------+ | |
|+> ---| H |--| RZ(2a_1) |--+--| ZZ_01| *-----------*-----------
+---+ +----------+ +------+ | +------+ |
+---+ +----------+ | | | | +------+
|+> ---| H |--| RZ(2a_2) |--------------+--| ZZ_02| *--| ZZ_12|--
+---+ +----------+ +------+ | +------+
+---+ +----------+ | +------+
|+> ---| H |--| RZ(2a_3) |--------------------------+--| ZZ_03|--
+---+ +----------+ +------+
(+ ZZ_13, ZZ_23)
a_i = time_step * x_i (single-qubit angles)
ZZ_ij uses angle = time_step * (pi - x_i)(pi - x_j)
3.2 XY Model¶
Non-diagonal Hamiltonian with XX and YY interactions. Creates coherent superpositions through off-diagonal coupling --- enables "hopping" of excitations between qubits.
3.3 Heisenberg Model¶
The most expressive type. Combines XX, YY, and ZZ interactions --- the full isotropic spin-½ Heisenberg Hamiltonian.
H_heis(x) = [SUM_i x_i Z_i] + SUM_{i<j} f(x_i, x_j) (X_i X_j + Y_i Y_j + Z_i Z_j)
optional XX + YY + ZZ interactions
3.4 Pauli-Z¶
Simplest entangling type. Only Z and ZZ terms (identical to IQP in structure but distinguished for API clarity).
Comparison at a Glance¶
+==============+============+============+===================+==============+
| Type | Single-q | Two-qubit | Interaction | Expressivity |
| | Terms | Terms | Character | |
+==============+============+============+===================+==============+
| iqp | Z_i | ZZ | Diagonal (phase) | High |
| pauli_z | Z_i | ZZ | Diagonal (phase) | High |
| xy | Z_i (opt) | XX + YY | Off-diagonal | Higher |
| heisenberg | Z_i (opt) | XX+YY+ZZ | Full isotropic | Highest |
+==============+============+============+===================+==============+
4. Two-Qubit Gate Decompositions¶
Each Pauli interaction exp(-i * angle * PP) is decomposed into native
gates (CNOT, RZ, H, S/Sdg). This is how the abstract Hamiltonian becomes
a real circuit.
ZZ Interaction: exp(-i * angle * Z_i Z_j)¶
q_i ----*-------------------*----
| |
q_j ----+---[ RZ(2*angle) ]--+----
CNOT CNOT
Depth: 3 layers Gates: 2 CNOT + 1 RZ
XX Interaction: exp(-i * angle * X_i X_j)¶
q_i --[H]---*-------------------*---[H]--
| |
q_j --[H]---+---[ RZ(2*angle) ]--+---[H]--
CNOT CNOT
Depth: 5 layers Gates: 4 H + 2 CNOT + 1 RZ
Idea: H rotates X-basis -> Z-basis, apply ZZ, rotate back.
YY Interaction: exp(-i * angle * Y_i Y_j)¶
q_i --[Sdg]--[H]---*-------------------*---[H]--[S]--
| |
q_j --[Sdg]--[H]---+---[ RZ(2*angle) ]--+---[H]--[S]--
CNOT CNOT
Depth: 7 layers Gates: 2 Sdg + 4 H + 2 CNOT + 1 RZ + 2 S
Idea: Sdg rotates Y-basis -> X-basis, H -> Z-basis, apply ZZ, reverse.
5. Rotation Angle Formulas¶
Single-Qubit Terms¶
Linear in feature value. Larger features produce larger rotations.
Two-Qubit Terms (Quantum Kernel Feature Map)¶
This formula comes directly from the quantum kernel literature (Havlicek et al., 2019). It has a critical point at x = pi where interactions vanish:
Interaction Strength vs Feature Value
(for x_i = x_j = x, time_step = 1)
Strength
|
|**
| **
| **
| ***
| ****
| *****
| ********
| **********
+---|---|---|---|---|---|---|---|---|---> x
0 0.5 1 1.5 2 2.5 3 pi 3.5
Maximum at x=0: angle = pi^2 ~ 9.87
Zero at x=pi: angle = 0 (CRITICAL POINT)
WARNING: Features near pi (~3.14) produce near-zero interactions!
Recommended input ranges: [0, 1] or [-1, 1]
6. Entanglement Topologies¶
The entanglement parameter controls which qubit pairs interact.
This determines both circuit depth and hardware connectivity requirements.
Full Entanglement¶
Every pair of qubits interacts. Maximum expressivity, but O(n^2) gates.
n=5 qubits, full entanglement: C(5,2) = 10 pairs
q0 ----*----*----*----*----
| | | |
q1 ----+----|----|----|---------*----*----*----
| | | | | |
q2 ---------+----|----|---- +----|----|----*----*----
| | | | | |
q3 --------------+----|--------- +----|----|----+----
| | |
q4 -------------------+-------------------+----+--------
10 pairs: (0,1)(0,2)(0,3)(0,4)(1,2)(1,3)(1,4)(2,3)(2,4)(3,4)
Requires: all-to-all qubit connectivity
Linear Entanglement¶
Only nearest neighbors interact. Hardware-friendly, O(n) gates.
n=5 qubits, linear entanglement: 4 pairs
q0 ----*---------------------------------------------
|
q1 ----+----*----------------------------------------
|
q2 ---------+----*-----------------------------------
|
q3 --------------+----*------------------------------
|
q4 -------------------+------------------------------
4 pairs: (0,1)(1,2)(2,3)(3,4)
Requires: nearest-neighbor connectivity only
Circular Entanglement¶
Nearest neighbors plus wrap-around. Captures periodic boundary conditions.
n=5 qubits, circular entanglement: 5 pairs
q0 ----*---------------------------------------------*----
| |
q1 ----+----*-----------------------------------------|----
| |
q2 ---------+----*-----------------------------------|----
| |
q3 --------------+----*------------------------------|----
| |
q4 -------------------+------------------------------+----
wrap-around
5 pairs: (0,1)(1,2)(2,3)(3,4)(4,0)
Requires: ring topology connectivity
Topology Scaling¶
+===========+====================+==================+==================+
| Topology | Pairs | CNOT per | Connectivity |
| | | Trotter step | Required |
+===========+====================+==================+==================+
| full | n(n-1)/2 | O(n^2) | All-to-all |
| linear | n - 1 | O(n) | Nearest-neighbor |
| circular | n | O(n) | Ring |
+===========+====================+==================+==================+
n_features | full | linear | circular
-----------+--------+----------+-----------
4 | 6 | 3 | 4
8 | 28 | 7 | 8
16 | 120 | 15 | 16
32 | 496 | 31 | 32
64 | 2016 | 63 | 64
7. Complete Circuit Structure¶
The full circuit for one Trotter step with IQP type, 4 qubits, full entanglement:
Layer 1: Layer 2: Layer 3:
Initial Single-qubit Two-qubit interactions
superposition Z rotations (ZZ for each pair)
|0> --[H]--------[RZ(2a_0)]--------*---[ZZ_01]---*---[ZZ_02]---*---[ZZ_03]--
| | |
|0> --[H]--------[RZ(2a_1)]--------+-------------*---[ZZ_12]---*---[ZZ_13]--
| |
|0> --[H]--------[RZ(2a_2)]------------------------+-------------*---[ZZ_23]--
|
|0> --[H]--------[RZ(2a_3)]------------------------------------------+------
With reps=2 Trotter steps:
|0> --[H]--[RZ]--[ZZ pairs]--||--[RZ]--[ZZ pairs]--
|0> --[H]--[RZ]--[ZZ pairs]--||--[RZ]--[ZZ pairs]-- || = barrier
|0> --[H]--[RZ]--[ZZ pairs]--||--[RZ]--[ZZ pairs]--
|0> --[H]--[RZ]--[ZZ pairs]--||--[RZ]--[ZZ pairs]--
^ ^
Trotter step 1 Trotter step 2
For Heisenberg type, each pair gets XX + YY + ZZ:
Each pair (i,j) in one Trotter step:
q_i --[RZ]--[H]*[CNOT]*[RZ]*[CNOT]*[H]--[Sdg][H]*[CNOT]*[RZ]*[CNOT]*[H][S]--*[CNOT]*[RZ]*[CNOT]*--
q_j --------[H]+------+[RZ]+------+[H]--[Sdg][H]+------+[RZ]+------+[H][S]--+------+[RZ]+------+--
^--- XX interaction ---^ ^---------- YY interaction ----------^ ^-- ZZ interaction --^
8. Key Properties¶
+==========================================================================+
| HAMILTONIAN ENCODING PROPERTIES |
+============================+=============================================+
| Qubits required | n (one per feature) |
| Circuit depth | 1 + reps * (interaction depth per type) |
| Initial state | |+>^n (Hadamard layer) |
| Entangling? | YES (two-qubit gates create entanglement)|
| Simulability | Not classically simulable |
| Trainable parameters | 0 (all data-dependent) |
| Trainability | Depends on depth and entanglement |
| Connectivity required | Depends on topology (none to all-to-all) |
+============================+=============================================+
Gate Count by Hamiltonian Type (per Trotter step, p = number of pairs)¶
+==============+=========+=======+=====+========+=======+
| Type | H | RZ | S | CNOT | Total |
+==============+=========+=======+=====+========+=======+
| iqp/pauli_z | n (init)| n + p | 0 | 2p | |
| xy | n + 8p | [n]+2p| 4p | 4p | |
| heisenberg | n + 8p | [n]+3p| 4p | 6p | |
+==============+=========+=======+=====+========+=======+
[n] = if include_single_qubit_terms
Where: n = n_qubits, p = n_entanglement_pairs
For n=4 qubits, full entanglement (p=6), reps=1:
+--------------+-----+-----+----+------+-------+
| Type | H | RZ | S | CNOT | Total |
+--------------+-----+-----+----+------+-------+
| iqp | 4 | 10 | 0 | 12 | 26 |
| xy | 52 | 16 | 24 | 24 | 116 |
| heisenberg | 52 | 22 | 24 | 36 | 134 |
+--------------+-----+-----+----+------+-------+
9. Depth Analysis and Parallelization¶
Two-qubit gates on disjoint qubit pairs can execute in parallel. The parallelization depends on the entanglement topology and maps to the edge coloring problem from graph theory.
Topology Graph Chromatic Index Parallel Layers
========= =========== ================= ===============
linear Path P_n 2 2
circular Cycle C_n 2 (n even) / 3 (n odd) 2 or 3
full Complete K_n n-1 (n even) / n (n odd) n-1 or n
Example: Linear, n=6¶
Pairs: (0,1) (1,2) (2,3) (3,4) (4,5)
Group A (disjoint): (0,1) (2,3) (4,5) <-- can run in parallel
Group B (disjoint): (1,2) (3,4) <-- can run in parallel
=> 2 parallel layers instead of 5 sequential
Depth per Trotter Step¶
+============+============================+================================+
| Type | Depth per parallel group | Total depth per Trotter step |
+============+============================+================================+
| iqp/pz | 1(RZ) + 3(ZZ) * groups | 1 + 3*G |
| xy | [1(RZ)] + 5(XX)*G + 7(YY)*G | [1] + 12*G |
| heisenberg | [1(RZ)] + 5(XX)*G + 7(YY)*G + 3(ZZ)*G | [1] + 15*G |
+============+============================+================================+
G = number of parallel groups
Total depth = 1 (Hadamard) + reps * (depth per Trotter step)
10. Parameters Guide¶
+=============================+==========+==========================================+
| Parameter | Default | Effect |
+=============================+==========+==========================================+
| | | |
| n_features | required | Number of features = number of qubits. |
| | | |
| hamiltonian_type | 'iqp' | Interaction structure: |
| | | 'iqp' - ZZ (diagonal) |
| | | 'pauli_z' - ZZ (same as iqp) |
| | | 'xy' - XX + YY |
| | | 'heisenberg' - XX + YY + ZZ |
| | | |
| evolution_time | 1.0 | Total time t in exp(-iHt). |
| | | Larger = stronger encoding. |
| | | Too large = Trotter error. |
| | | |
| reps | 2 | Number of Trotter steps. |
| | | More = better approximation, |
| | | deeper circuit. |
| | | |
| entanglement | 'full' | Qubit pair topology: |
| | | 'full' - all pairs, O(n^2) |
| | | 'linear' - nearest-neighbor, O(n) |
| | | 'circular' - ring, O(n) |
| | | |
| include_single_qubit_terms | True | Add Z_i single-qubit rotations. |
| | | Always True for iqp/pauli_z. |
| | | Optional for xy/heisenberg. |
| | | |
| max_pairs | None | Limit number of qubit pairs. |
| | | Truncates full entanglement for |
| | | large n to control depth. |
| | | |
| insert_barriers | True | Barriers between Trotter steps |
| | | (visualization, prevent optimization). |
| | | |
+=============================+==========+==========================================+
11. Example Walkthrough¶
Encode x = [0.3, 0.7, 0.1] with IQP Hamiltonian, evolution_time=1.0,
reps=1, entanglement='full':
Step 1: Compute time step
-------------------------
time_step = evolution_time / reps = 1.0 / 1 = 1.0
Step 2: Compute single-qubit angles
------------------------------------
a_0 = time_step * x_0 = 1.0 * 0.3 = 0.30
a_1 = time_step * x_1 = 1.0 * 0.7 = 0.70
a_2 = time_step * x_2 = 1.0 * 0.1 = 0.10
Step 3: Compute two-qubit angles (formula: t * (pi - x_i)(pi - x_j))
---------------------------------------------------------------------
a_01 = 1.0 * (pi - 0.3)(pi - 0.7) = 2.841 * 2.441 = 6.935
a_02 = 1.0 * (pi - 0.3)(pi - 0.1) = 2.841 * 3.041 = 8.640
a_12 = 1.0 * (pi - 0.7)(pi - 0.1) = 2.441 * 3.041 = 7.424
Step 4: Build circuit
---------------------
|0> --[H]--[RZ(0.60)]--*--[CNOT]--[RZ(13.87)]--[CNOT]--*-----------..
| |
|0> --[H]--[RZ(1.40)]--+--------------------------------+--*--------..
|
|0> --[H]--[RZ(0.20)]--------------------------------------+--------..
(continues with ZZ_02 and ZZ_12 interactions)
Note: RZ angles are 2*a for the gate decomposition.
Step 5: Result
--------------
The output state |psi(x)> is an ENTANGLED 3-qubit state.
Unlike product-state encodings, individual qubits are correlated.
Measurement outcomes exhibit quantum correlations from ZZ coupling.
12. Trotterization: Accuracy vs Depth¶
The Trotter approximation introduces error that decreases with more steps. This is the fundamental tradeoff of Hamiltonian encoding.
Trotter Error ~ O( t^2 / r )
where: t = evolution_time, r = reps
For evolution_time=1.0:
reps | Error bound | Circuit depth multiplier
-----+--------------+---------------------------
1 | O(1) | 1x (minimum depth)
2 | O(0.25) | 2x
4 | O(0.0625) | 4x
8 | O(0.0156) | 8x
16 | O(0.0039) | 16x
In practice, reps=2 is sufficient for most ML tasks.
Exact Trotter accuracy is less critical than for physics simulation
because the ML model learns to use the encoded representation
regardless of small approximation errors.
Fidelity with exact evolution:
reps=1 ||||||||||||________________ ~0.6 (rough approximation)
reps=2 ||||||||||||||||||||________ ~0.85 (good for ML)
reps=4 ||||||||||||||||||||||||||__ ~0.95 (high fidelity)
reps=8 |||||||||||||||||||||||||||| ~0.99 (near-exact)
13. Comparison with Related Encodings¶
+===================+==========+=========+============+===============+
| Encoding | Qubits | Depth | Entangling | Approach |
+===================+==========+=========+============+===============+
| Hamiltonian * | n | O(rn^2) | YES | Time evolution|
| IQP Encoding | n | O(rn^2) | YES | Direct ZZ |
| ZZ Feature Map | n | O(rn^2) | YES | Direct ZZ |
| Pauli Feature Map| n | O(rn^2) | YES | Pauli rotations|
| Angle | n | O(r) | NO | Direct angles |
| Higher-Order | n | O(r) | NO | Poly angles |
| Basis | n | O(1) | NO | Bit strings |
| Amplitude | log_2(n) | O(2^n) | YES | State prep |
+===================+==========+=========+============+===============+
* Hamiltonian encoding is distinguished by its PHYSICS-MOTIVATED design.
It doesn't just apply gates --- it simulates a physical system.
What Makes Hamiltonian Encoding Special?¶
+-------------------------------------------------------------------+
| Feature | IQP/ZZ/Pauli Maps | Hamiltonian Encoding |
+-----------------------+--------------------+----------------------+
| Gate application | Direct, ad-hoc | From Trotterized |
| | | time evolution |
| | | |
| evolution_time | No analogue | Explicit control |
| parameter | | over encoding strength|
| | | |
| Multiple interaction | Choose one | Choose from IQP, XY, |
| types | (ZZ only) | Heisenberg, Pauli-Z |
| | | |
| Physical motivation | Circuit-centric | Hamiltonian dynamics |
| | | (quantum simulation) |
| | | |
| Trotter framework | Fixed structure | Systematic accuracy |
| | | improvement via reps |
+-------------------------------------------------------------------+
14. Data Preprocessing¶
The two-qubit angle formula (pi - x_i)(pi - x_j) makes input scaling
especially important for Hamiltonian encoding.
+==================+=================================+=======================+
| Input Range | Two-qubit angle range | Recommendation |
+==================+=================================+=======================+
| [0, 1] | [4.60, 9.87] * time_step | SAFE (recommended) |
| [-1, 1] | [4.60, 17.2] * time_step | SAFE (wide range) |
| [0, pi] | [0, 9.87] * time_step | CAUTION (includes 0) |
| [0, 2*pi] | includes critical point at pi | AVOID |
+==================+=================================+=======================+
Critical point: When x_i = pi, the factor (pi - x_i) = 0.
This makes the two-qubit interaction VANISH.
Recommended: Normalize features to [0, 1] or [-1, 1] before encoding.
15. Strengths and Limitations¶
STRENGTHS LIMITATIONS
+-----------------------------+ +-------------------------------+
| | | |
| + Physics-grounded design | | - O(n^2) gates for full |
| Hamiltonian dynamics | | entanglement (quadratic) |
| provide natural inductive| | |
| bias | | - Critical point at x = pi |
| | | requires careful input |
| + Rich entanglement | | scaling |
| Creates highly entangled | | |
| states via two-qubit | | - Trotter error with few |
| interactions | | reps (accuracy vs depth) |
| | | |
| + Flexible Hamiltonian | | - Deep circuits may exhibit |
| Four types to match | | barren plateaus in |
| different problems | | gradient-based training |
| | | |
| + Explicit evolution_time | | - Full entanglement needs |
| Fine-grained control | | all-to-all connectivity |
| over encoding strength | | (not always available) |
| | | |
| + Not classically simulable| | - Heisenberg type has |
| Potential for quantum | | highest gate overhead |
| advantage in kernels | | (6 CNOT per pair per rep) |
| | | |
+-----------------------------+ +-------------------------------+
16. When to Use Hamiltonian Encoding¶
Best suited for
+----------------------------+
| |
+---------------+ Quantum Kernel Methods | Physics-motivated kernels
| | and Classification | with configurable structure
| +----------------------------+
| | |
+---------------+ Physics-Inspired ML | Molecular properties,
| | Problems | materials science,
| +----------------------------+ quantum chemistry
| | |
+---------------+ Exploring Hamiltonian | Systematic comparison of
| | Structures for ML | IQP vs XY vs Heisenberg
| +----------------------------+ for your specific task
| | |
+---------------+ Hardware with Good | Leverage entanglement
| | Two-Qubit Fidelity | when hardware supports it
| +----------------------------+
| | |
+---------------+ Time-Series Data | Natural temporal structure
| Encoding | via evolution_time
+----------------------------+
17. Configuration Quick Reference¶
+============================================+===================================+
| USE CASE | RECOMMENDED CONFIG |
+============================================+===================================+
| Standard quantum kernel (Havlicek 2019) | hamiltonian_type='iqp', |
| | entanglement='full' |
| | |
| NISQ-friendly (limited connectivity) | entanglement='linear', reps=2 |
| | |
| Maximum expressivity | hamiltonian_type='heisenberg', |
| | entanglement='full' |
| | |
| Coherent excitation transfer | hamiltonian_type='xy' |
| | |
| Large feature count (n > 20) | entanglement='linear' or |
| | max_pairs=100 |
| | |
| Periodic boundary conditions | entanglement='circular' |
| | |
| High Trotter fidelity | reps=4+, evolution_time=1.0 |
| | |
| Strong encoding (wider rotations) | evolution_time=2.0 or higher |
| | (increase reps proportionally) |
+============================================+===================================+
18. Resource Scaling¶
IQP type, full entanglement, reps=2 (initial H layer counted once):
n_features | Pairs | H gates | RZ gates | CNOT gates | Total | Depth
-----------+-------+---------+----------+------------+-------+------
2 | 1 | 2 | 6 | 4 | 12 | 7
4 | 6 | 4 | 20 | 24 | 48 | 25
6 | 15 | 6 | 42 | 60 | 108 | 31
8 | 28 | 8 | 72 | 112 | 192 | 43
10 | 45 | 10 | 110 | 180 | 300 | 55
16 | 120 | 16 | 272 | 480 | 768 | 91
Linear entanglement, reps=2 (same n values):
n_features | Pairs | CNOT gates | Total | Depth
-----------+-------+------------+-------+------
4 | 3 | 12 | 30 | 13
8 | 7 | 28 | 66 | 13
16 | 15 | 60 | 136 | 13
32 | 31 | 124 | 272 | 13
64 | 63 | 252 | 544 | 13
Key: linear entanglement has CONSTANT depth (independent of n)
because all nearest-neighbor pairs fit in 2 parallel groups!
References¶
-
Schuld, M., Sweke, R., & Meyer, J. J. (2021). "Effect of data encoding on the expressive power of variational quantum-machine-learning models." Physical Review A, 103(3), 032430.
-
Havlíček, V., et al. (2019). "Supervised learning with quantum-enhanced feature spaces." Nature, 567(7747), 209-212.