# Simulating the Deutsch–Jozsa algorithm at the pulse level¶

Author: Boxi Li (etamin1201@gmail.com)

In this example, we demonstrate how to simulate simple quantum algorithms on a qauntum hardware with QuTiP. The simulators are defined in the class Processor(and its sub-classes). Processor represents a general quantum device. The interaction of the quantum systems such as qubits is defined by the control Hamiltonian. For a general introduction of pulse-level simulation, please refer to the user guide.

In the following, we compile a simple three-qubit quantum circuit into control pulses on different Hamiltonian model.

## The Deutsch–Jozsa algorithm¶

The Deutsch–Jozsa algorithm is the simplest quantum algorithm that offers an exponential speed-up compared to the classical one. It assumes that we have a function $f:\{0,1\}^n \rightarrow \{0,1\}$ which is either balanced or constant. Constant means that $f(x)$ is either 1 or 0 for all inputs while balanced means that $f(x)$ is 1 for half of the input domain and 0 for the other half. A more rigorous definition can be found at https://en.wikipedia.org/wiki/Deutsch-Jozsa_algorithm.

The implementation of the Deutsch–Jozsa algorithm includes $n$ input qubits and 1 ancilla initialised in state $1$. At the end of the algorithm, the first $n$ qubits are measured on the computational basis. If the function is constant, the result will be $0$ for all $n$ qubits. If balanced, $\left|00...0\right\rangle$ will never be measured. The following example is implemented for the balanced function $f:\{00,01,10,11\} \rightarrow \{0,1\}$, where $f(00)=f(11)=0$ and $f(01)=f(10)=1$. This function is balanced, so the probability of measuring state $\left|00\right\rangle$ should be 0.

In [1]:
import numpy as np
from qutip_qip.device import OptPulseProcessor, LinearSpinChain, SpinChainModel, SCQubits
from qutip_qip.circuit import QubitCircuit
from qutip import sigmaz, sigmax, identity, tensor, basis, ptrace

In [2]:
qc = QubitCircuit(N=3)

# function f(x)


In [3]:
qc

Out[3]:

## Using the spin chain model¶

First, we simulate the quantum circuit using the Hamiltonian model LinearSpinChain. The control Hamiltonians are defined in SpinChainModel.

In [4]:
processor = LinearSpinChain(3)


To quickly visualize the pulse, Processor has a method called plot_pulses. In the figure bellow, each colour represents the pulse sequence of one control Hamiltonian in the system as a function of time. In each time interval, the pulse remains constant.

In [5]:
processor.plot_pulses(title="Control pulse of Spin chain", figsize=(8, 4), dpi=100);


Because for the spin chain model interaction only exists between neighbouring qubits, SWAP gates are added between and after the first CNOT gate, swapping the first two qubits. The SWAP gate is decomposed into three iSWAP gates, while the CNOT is decomposed into two iSWAP gates plus additional single-qubit corrections. Both the Hadamard gate and the two-qubit gates need to be decomposed to native gates (iSWAP and rotation on the $x$ and $z$ axes). The compiled coefficients are square pulses and the control coefficients on $\sigma_z$ and $\sigma_x$ are also different, resulting in different gate times.

### Without decoherence¶

In [6]:
basis00 = basis([2,2], [0,0])
psi0 = basis([2,2,2], [0,0,0])
result = processor.run_state(init_state=psi0)
print("Probability of measuring state 00:")
print(np.real((basis00.dag() * ptrace(result.states[-1], [0,1]) * basis00)[0,0]))

Probability of measuring state 00:
1.543983026671485e-08


### With decoherence¶

In [7]:
processor.t1 = 100
processor.t2 = 30
psi0 = basis([2,2,2], [0,0,0])
result = processor.run_state(init_state=psi0)
print("Probability of measuring state 00:")
print(np.real((basis00.dag() * ptrace(result.states[-1], [0,1]) * basis00)[0,0]))

Probability of measuring state 00:
0.13730255188167628


## Using the optimal control module¶

This feature integrated into the sub-class OptPulseProcessor which use methods in the optimal control module to find the optimal pulse sequence for the desired gates. It can find the optimal pulse either for the whole unitary evolution or for each gate. Here we choose the second option.

In [8]:
setting_args = {"SNOT": {"num_tslots": 6, "evo_time": 2},
"X": {"num_tslots": 1, "evo_time": 0.5},
"CNOT": {"num_tslots": 12, "evo_time": 5}}
processor = OptPulseProcessor(  # Use the control Hamiltonians of the spin chain model.
num_qubits=3, model=SpinChainModel(3, setup="linear"))
processor.load_circuit(  # Provide parameters for the algorithm
qc, setting_args=setting_args, merge_gates=False,
verbose=True, amp_ubound=5, amp_lbound=0);

********** Gate 0 **********
Final fidelity error 6.6028293943531935e-12
Final gradient normal 0.0015194067969008138
Terminated due to Goal achieved
Number of iterations 6
********** Gate 1 **********
Final fidelity error 0.004418764518783647
Final gradient normal 0.05370962792587447
Terminated due to Iteration or fidelity function call limit reached
Number of iterations 500
********** Gate 2 **********
Final fidelity error 0.018942311916405496
Final gradient normal 0.029431698379522316
Terminated due to function converged
Number of iterations 478
********** Gate 3 **********
Final fidelity error 0.017966984837258115
Final gradient normal 0.06234535946891994
Terminated due to function converged
Number of iterations 160
********** Gate 4 **********
Final fidelity error 6.029392540796152e-09
Final gradient normal 8.129992922909755e-05
Terminated due to function converged
Number of iterations 129
********** Gate 5 **********
Final fidelity error 3.227445011244612e-08
Final gradient normal 0.0006879404036636063
Terminated due to function converged
Number of iterations 100
********** Gate 6 **********
Final fidelity error 0.008469231492828078
Final gradient normal 0.004922643478312693
Terminated due to function converged
Number of iterations 370
********** Gate 7 **********
Final fidelity error 0.024904080477859303
Final gradient normal 0.04025417713677587
Terminated due to function converged
Number of iterations 255

In [9]:
processor.plot_pulses(title="Control pulse of OptPulseProcessor", figsize=(8, 4), dpi=100);


For the optimal control model, we use the GRAPE algorithm, where control pulses are piece-wise constant functions. We provide the algorithm with the same control Hamiltonian model used for the spin chain model. In the compiled optimal signals, all controls are active (non-zero pulse amplitude) during most of the execution time. We note that for identical gates on different qubits (e.g., Hadamard), each optimized pulse is different, demonstrating that the optimized solution is not unique, and there are further constraints one could apply, such as adaptions for the specific hardware.

### Without decoherence¶

In [10]:
basis00 = basis([2,2], [0,0])
psi0 = basis([2,2,2], [0,0,0])
result = processor.run_state(init_state=psi0)
print("Probability of measuring state 00:")
print(np.real((basis00.dag() * ptrace(result.states[-1], [0,1]) * basis00)[0,0]))

Probability of measuring state 00:
0.06973417322091019


### With decoherence¶

In [11]:
processor.t1 = 100
processor.t2 = 30
psi0 = basis([2,2,2], [0,0,0])
result = processor.run_state(init_state=psi0)
print("Probability of measuring state 00:")
print(np.real((basis00.dag() * ptrace(result.states[-1], [0,1]) * basis00)[0,0]))

Probability of measuring state 00:
0.20122806768707008


We can see that under noisy evolution their is a none zero probability of measuring state 00.

## Using the superconducting qubits model¶

Below, we simulate the same quantum circuit using one sub-class LinearSpinChain. It will find the pulse based on the Hamiltonian available on a quantum computer of the linear spin chain system. Please refer to the notebook of the spin chain model for more details.

In [12]:
processor = SCQubits(num_qubits=3)

In [13]:
processor.plot_pulses(title="Control pulse of SCQubits", figsize=(8, 4), dpi=100);


For the superconducting-qubit processor, the compiled pulses have a Gaussian shape. This is crucial for superconducting qubits because the second excited level is only slightly detuned from the qubit transition energy. A smooth pulse usually prevents leakage to the non-computational subspace. Similar to the spin chain, SWAP gates are added to switch the zeroth and first qubit and one SWAP gate is compiled to three CNOT gates. The control $ZX^{21}$ is not used because there is no CNOT gate that is controlled by the second qubit and acts on the first one.

### Without decoherence¶

In [14]:
basis00 = basis([3, 3], [0, 0])
psi0 = basis([3, 3, 3], [0, 0, 0])
result = processor.run_state(init_state=psi0)
print("Probability of measuring state 00:")
print(np.real((basis00.dag() * ptrace(result.states[-1], [0,1]) * basis00)[0,0]))

Probability of measuring state 00:
0.00034459284476520447


### With decoherence¶

In [15]:
processor.t1 = 50.e3
processor.t2 = 20.e3
psi0 = basis([3, 3, 3], [0, 0, 0])
result = processor.run_state(init_state=psi0)
print("Probability of measuring state 00:")
print(np.real((basis00.dag() * ptrace(result.states[-1], [0,1]) * basis00)[0,0]))

Probability of measuring state 00:
0.060418994841937905

In [16]:
import qutip_qip
print("qutip-qip version:", qutip_qip.version.version)
from qutip.ipynbtools import version_table
version_table()

qutip-qip version: 0.2.0

Out[16]:
SoftwareVersion
QuTiP4.6.3
Numpy1.22.2
SciPy1.8.0
matplotlib3.5.1
Cython0.29.27
Number of CPUs12
BLAS InfoOPENBLAS
IPython8.0.1
Python3.9.0 | packaged by conda-forge | (default, Nov 26 2020, 07:53:15) [MSC v.1916 64 bit (AMD64)]
OSnt [win32]
Thu Feb 10 20:43:57 2022 W. Europe Standard Time
In [ ]: