Quantum Refrigerator
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Reference
Physical limitations¶
Verify the following statement on page 10 of the paper:
A refrigerator with these properties (i.e., outputs \(\ket0\) with error bounded by constant \({\color{red}\epsilon_2}\)) can be achieved with a constant value of \(R\) and a quantum circuit of constant size \(F\).
Short answer: Yes, if by "constant" the authors are referring to the number of qubits required for performing FTQC (Fault-Tolerant Quantum Computing) — or \(n'\) in the paper's context.
Otherwise, the values of \(R\) and \(F\) may not be constant if considered with respect to the error bound \(\color{red}{\epsilon_2}\).
(But generally I think what the authors are trying to say is that the fridges' size would remain the same if the FTQC scales up.)
What affects \(R\) and \(F\)?
- \(R\) --> the qubit number that each fridge takes as input.
After cooling, each fridge outputs \(\color{blue}{1}\) "fresh" qubit (almost in state \(\ket0\bra0\)), plus \(\color{blue}{R-1}\) "dirty" qubits (almost in maximally mixed state). - \(F\) --> the depth of the fridge's circuit (for cooling purpose).
For instance, \(R=6\) in this simple demo of algorithmic cooling.
As for \(F\), the previous paper (the ref in algo-cooling) did not calculate how it scales with \(R\) or \(\epsilon_2\). Maybe I can try to calculate its lower/upper bounds.
The authors state in the paper:
(\(R\) and \(F\) will in general depend on \(P\) , or at least on \(S(P)\).)
I believe this is true. \(P\) is the "fixed point" of that non-unital channel. For example the amplitude-damping channel's \(P\) is just the North pole of the Bloch sphere. Since the puspose of cooling is to decrease entropy, it would help a lot if we start from a state which already has a low entropy, i.e., \(F\) will in general depend on \(S(P)\).
Full-Sketch of the Scheme¶
Noise Channels in the Fridge¶
| Noise Channel Class | Fixed Point(s) | Impact on Entropy | Computation Limit (given \(n\) physical qubits) | Example Channel | Key Features |
|---|---|---|---|---|---|
| Depolarizing Class | Center of the Bloch Sphere | Increases entropy for all states, eventually reaching maximum entropy (completely mixed state) | \(\tilde{O}(\log(n))\) time steps | Depolarizing Channel | - Drives all states towards maximum entropy, limiting computation time due to the difficulty in maintaining fresh ancilla qubits. |
| Dephasing Class | A diameter of the Bloch Sphere | Entropy does not decrease; it either increases for some states or remains constant for others. | \(O(n^b)\) time steps with \(O(n^a)\) qubits (if \(a+b<1\)) | Dephasing Channel | - Allows for polynomial-time computations by using a subset of qubits as fresh ancilla sources. - Limits quantum information storage time due to the inability to reduce entropy. |
 Non-Unital Class |
A single point (not the Bloch center) | Allows for entropy to decrease under the channel, enabling "cooling" of qubits. | Exponential number of time steps possible |
Amplitude Damping Channel | - Enables exponentially long computations due to the inherent "cooling" property, allowing ancilla qubit "refrigeration". |
Note: The computation time for the "Non-unital Class" while stated as "exponential" in the source, is noted as a likely upper limit rather than a definitive bound.
Some Draft Notes¶
The main theorem establishes the feasibility of quantum computation under various noise models, categorized by their impact on computation time.
Theorem 1.1 (Main Theorem)
Let \(C\) be any non-unitary channel close to the identity, and consider quantum computations which suffer from noise \(C\) on each qubit at each time step. Up to unitary equivalence, the limit of repeatedly applying \(C\) can be either a point or a diameter in the Bloch sphere.
- If the limit is the center of the Bloch sphere, it is possible to compute for \(\tilde{O}(\log(n))\) time steps. This is tight up to \(\log\log n\) factors. An example of this class is the depolarizing channel.
- If the limit is a diameter, then it is possible to compute on \(O(n^a)\) qubits for \(O(n^b)\) time steps provided \(a + b < 1\), and impossible even to store a single (unknown) qubit for more than \(O(n^3)\) time steps. An example of this class is the dephasing channel.
- If the limit is a point which is not the center of the Bloch sphere, it is possible to compute for an exponential number of time steps. An example of this class is the amplitude damping channel.
Explanation:
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Channel Classification: The theorem classifies quantum channels based on the long-term behavior of qubits subjected to the channel. This behavior is characterized by the "fixed point" of the channel, represented as a point or a diameter on the Bloch sphere (a geometrical representation of a qubit's state).
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Case 1 (Center of Bloch Sphere): When the channel drives all states towards the maximally mixed state (center of the Bloch sphere), computation time is limited to a logarithmic scale (\(\tilde{O}(\log(n))\)). The depolarizing channel, which mixes qubit states, exemplifies this class. This limitation arises from the channel's effect of increasing entropy, making it challenging to maintain low-entropy ancilla qubits needed for error correction over extended periods.
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Case 2 (Diameter of Bloch Sphere): Channels in this class contract the Bloch sphere towards a diameter. For instance, the dephasing channel, drawing states towards the Z-axis, falls into this category. The theorem states that computations are possible in this scenario, but with a polynomial trade-off between the number of qubits used (\(O(n^a)\)) and the achievable computation time (\(O(n^b)\)), constrained by \(a + b < 1\). This limitation originates from the channel's inability to decrease entropy, eventually leading to information loss. However, storing quantum information faces a stricter limitation (\(O(n^3)\) time steps).
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Case 3 (Point other than Center): This class comprises non-unital channels, characterized by a unique fixed point that isn't the center of the Bloch sphere. The amplitude damping channel, driving states to a specific point on the Z-axis, illustrates this case. Notably, these channels allow for entropy decrease, offering a "cooling" mechanism. The theorem demonstrates that this cooling facilitates exponentially long computations using \(O(n \text{ polylog}(nD))\) physical qubits for a computation of width \(n\) and depth \(D\). This significant improvement stems from the ability to "refrigerate" and reuse ancilla qubits, removing the dependence on an external supply of fresh ones.
Significance: This theorem highlights the crucial link between a quantum channel's fixed points and its capacity to support long computations. The ability to reduce entropy, a unique feature of non-unital channels, emerges as a key factor in enabling robust and scalable quantum computation.
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