Awesome

 Just had to do it: I ran the Max-Cut tutorial from Cirq-google. It was easy

to set up; one can download the notebook and it's just copy-paste to Jupyter

Notebook. Ran without a glitch...

Reading through it, I noticed it seemed to be suited to max-cut anything that is

a subset of a larger qubit ensemble. So I asked for a 5-node one. The result:

Awesome...

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Chat with MERLIN(Chatgpt3.5):

quantum computing; how does a negative cost hamiltonian come about?

In quantum mechanics, a Hamiltonian represents the total energy of a system, and can be expressed in terms of operators acting on quantum states. A negative cost Hamiltonian specifically refers to situations where the eigenvalues of the Hamiltonian associated with certain states can take on negative values. This concept is often discussed in the context of quantum computing and quantum algorithms, particularly those involving optimization problems or variational principles. Here’s how a negative cost Hamiltonian comes about:

 1. Context in Quantum Computing:

A Hamiltonian can be designed to encode the cost or quality of solutions to optimization problems. In many scenarios, particularly in quantum annealing and variational quantum algorithms (like the Variational Quantum Eigensolver), the goal is to find the minimum energy state, corresponding to the optimal solution of the problem.

 2. Construction of Hamiltonians:

- Standard Optimization Problems: In classical optimization, cost functions often have non-negative values. However, in quantum frameworks, it’s common to define a Hamiltonian where energy levels lower than zero can correspond to better solutions. This framing can facilitate exploration of the solution space.

- Using Energy Landscapes: By defining a Hamiltonian where certain configurations of a quantum state have negative energy, you can encourage the system to evolve towards these configurations during a quantum computation. The Hamiltonian might include terms that penalize certain configurations (leading to a higher energy) and reward others with lower energy (even negative).

 3. Variational Settings:

In variational quantum algorithms, the Hamiltonian is often parameterized by variables that represent the state of the quantum system. This allows the energy levels to adjust based on the trial states:

- The ground state of a Hamiltonian that represents optimization can be associated with a negative value, signaling a higher probability of a correct solution as the state evolves towards the minimum energy.

 4. Quantum Annealing:

In quantum annealing, a well-known optimization strategy, the Hamiltonian mirrors the classical cost function being minimized. As the quantum system evolves (typically starting from a simple system with a known ground state), the Hamiltonian is slowly transformed. If the initial Hamiltonian incorporates a large positive offset, it might imply the actual optimizable landscape has regions of negative eigenvalues, effectively creating a landscape where good solutions reside at lower energies.

 5. Quantum Simulation:

Certain quantum simulations may leverage effective Hamiltonians that exhibit negative eigenvalues intentionally to allow simulations of phenomena with favorable properties or to enable certain types of interactions that are critical to finding optimal solutions.

 6. Physical Realizations:

Negative cost Hamiltonians can also arise from physical systems where interactions inherently lead to bound states, attractive forces, or specific potential shapes that allow for negative total energies in a bounded system.

 Conclusion:

In summary, a negative cost Hamiltonian emerges mainly from the goal of optimizing certain quantum systems and leveraging the unique probabilistic nature of quantum mechanics to find states corresponding to lower energy levels that represent optimal solutions. This is implemented through careful construction of Hamiltonians based on variational principles and quantum annealing techniques. 

Are there negative energy states within quantum mechanics itself?

Yes, negative energy states do exist in quantum mechanics, and they can be encountered in several contexts. Below are a few key aspects and examples where negative energy states are relevant:

 1. Bound States in Quantum Mechanics:

   - Potential Wells: In systems described by potential wells (like an electron in a quantum well or an atom), bound states can have negative energy. The energy of these states is often defined relative to the potential energy at infinity. For example, in a hydrogen atom, the electron’s energy levels are quantized and typically negative, meaning that the electron is bound to the nucleus. The ground state of a hydrogen atom, for instance, has an energy of approximately -13.6 eV.

   - Stability: The negative energy indicates that the bound state is preferred over a free state (where the energy would be zero or positive), reflecting a stable configuration where the particle is confined.

 2. Quantum Field Theory:

   - Vacuum States: In quantum field theory, the vacuum state can exhibit negative energy states due to the presence of virtual particles and fluctuations in the quantum fields. The concept of "zero-point energy" leads to the realization that even the vacuum has a non-zero energy, and interactions may result in negative effective mass or energy under certain conditions.

 3. Quantum Harmonic Oscillator:

   - In the quantum harmonic oscillator, the ground state energy is given byE₀ = 1/2 ℏw, which is positive. However, when one considers energy levels relative to other potentials or in interactions, effective negative energy states can be discussed in perturbation theories or in the context of coherent states.

 4. Superposition and Effective Hamiltonians:

   - In certain quantum systems, the effective Hamiltonian can exhibit negative eigenvalues when considering collective effects or certain interactions. This can especially be true in many-body systems, where interactions can lead to effective potentials that sink below zero.

 5. Casimir Effect:

   - The Casimir effect provides another example of negative energy density due to quantum fluctuations in the vacuum between two uncharged plates. The energy between the plates can become negative relative to the energy density in free space, illustrating how negative energy states emerge from quantum field effects.

 6. Quantum Gravity and Exotic States:

   - Although still theoretical, concepts in quantum gravity and string theory suggest that under certain circumstances, configurations may lead to negative energy states. These scenarios often involve exotic matter or hypothetical constructs but help illustrate that negative energy can be a consequential idea in advanced theoretical frameworks.

 Conclusion:

Negative energy states play a significant role in various branches of quantum mechanics and quantum field theory, ranging from bound states in atomic systems to vacuum fluctuations in quantum fields. Understanding these states is crucial in areas such as atomic physics, quantum information, and theoretical physics. They underscore the complex and often counterintuitive nature of quantum systems where the classical interpretation of energy does not always directly apply. 

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The cirq procedure:

Start with a ready-made qubit tray:                                                                                         

Choose a subset; here the first 12 qubits:

Define a circuit with 12 qubits:

Keep track of two variables:

Find the sensitivity to each variable:

Sort the qubits between high and low:

The answer...