QSwQH:
The Board Game

Dear QSwQH participants

I brought with me to Aspen this mind-bending quantum board game. and was hoping to play it with Aspen QSwQH participants during the break.  I have left the board with Andrew Daley, who is looking for people to play with.  The rules of the game are fairly straightforward.  But I already warned you--it is mind-bending, and you must discover the strategies for winning.

--Jeremy Levy

Overall goal

To advance one of your qubits from one side of the board to the other side using qugates.  The board is an 8x8 grid (same as a chess board).  The game is “optimized” for four players, but can be played with two players (but not three).

Overview

Each player starts with eight “Bloch Cubes” with a distinctive color (red, blue, green, yellow).

The role of each Bloch Cube depends on whether it is placed on the board or left off the board.

Bloch Cubes which are on the board (“in play”) are qubits which are constrained to be in one of six face-up states.  In addition, there are four possible phases for each state that is facing up, identified by the orientation of the “ket” or “.  The qubits must be aligned with the squares of the playing board except when “moving”.

Bloch Cubes not placed on the board function as “qugates”, and are used for manipulating the qubits that are on the board.  Each qugate performs a  (unitary) rotation of a qubit along the qugate axis that is face-up, using the right-hand rule.  The qubit on the board will either “roll” horizontally or vertically one square (if the qubit axis differs from the qugate axis) or “spins”  without changing location (if the qubit state matches the qugate axis).

Description of one player’s first turn:

Players initially roll all eight Bloch Cubes to randomize their state. The first move must be to place a Bloch Cube on the board, turning it into a qubit.  The initial phase of the qubit can be chosen by the player, but the state cannot change.  The remaining seven Bloch Cubes can either be used as qugates to operate on their qubits which are on the board, or placed on the board as well.  At the end of every turn, the player should roll their qugates to a new random configuration, to prepare for the next turn.

Strategy consideration

There is no limit to how many qubits can be placed on the board; however, the more qubits that are on the board, the fewer are available for gating.

Qugate operations, Rashba Spin-Orbit coupling, and example

A qugate either moves a qubit one square or spins it +-90 degrees.  For example, if the qubit is in the  face-up state with the right face in the  state, a qugate that is oriented with  face-up will “roll”  qubit forward one square (like Rashba spin-orbit coupling), and the new face-up state will be .   If the qubit is in the  state, then the  gate will spin the cube counterclockwise by 90 degrees.  It is possible for a qubit to leave the board, but only on the side where it originated.  In that case it is converted back into a qugate.  The action of a qugate on a qubit is also described in the two-minute video on the right column of this page.

Interactions between qubits: Pauli blocking and singlet states

Two qubits can only occupy a single square only if they are in a “singlet state”.

A singlet state is consists of a bottom qubit in  state and the top qubit in the  state.

The top qubit “traps” the bottom qubit, which is significant if the qubits belong to opposing players.

The singlet state is very powerful since you are allowed to perform a basis change at any time and convert

By convention, the top and bottom qubits of a singlet are “oriented” with ket-arrows facing opposite directions, and the overall rotation can also be chosen by the player who has the top qubit at any time.

Optional “measurement” rule

As a substitute for qugate action, a player may elect to use the  state of a given Bloch Cube to perform a “measurement” on a qubit.  The outcome of the measurement is determined by the Born rule.  If the qugate axis  matches the qubit axis , then nothing happens; otherwise, the qubit collapses randomly onto the -axis.  To determine the sign, the Bloch Cube is rolled.  A face-up value  determines a positive outcome so that the new state of the qubit is ;  if the Bloch Cube lands with a negative value  then the new state becomes .  To “collapse” the qubit to its measured state, it must be rotated from the side to the top with a single quarter-revolution motion, and the qubit does not change location.

You may choose to omit this rule if you are a believer in the Everett paradigm.

Conventions about how the game starts, proceeds, and ends

Someone should start somehow via Bloch Cube rolls determining who goes first.  

Play play should proceed counterclockwise if you are in the Northern hemisphere, and clockwise if you are in the Southern hemisphere.

Players should roll their qugates immediately after their turn.

Play continues until one player’s piece makes it across the board and off.

Other details

The Bloch Cube faces are labeled such that  and  states are on opposite sides, for

 values are arranged according to the right hand rule, with like signs sharing one corner.